Properties

Label 4-175e2-1.1-c1e2-0-1
Degree $4$
Conductor $30625$
Sign $1$
Analytic cond. $1.95267$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 2·4-s − 2·6-s − 2·7-s + 3·8-s + 2·9-s + 4·11-s − 4·12-s + 2·13-s + 2·14-s + 16-s + 4·17-s − 2·18-s − 4·21-s − 4·22-s − 8·23-s + 6·24-s − 2·26-s + 6·27-s + 4·28-s + 10·29-s − 6·31-s − 2·32-s + 8·33-s − 4·34-s − 4·36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 2/3·9-s + 1.20·11-s − 1.15·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 0.872·21-s − 0.852·22-s − 1.66·23-s + 1.22·24-s − 0.392·26-s + 1.15·27-s + 0.755·28-s + 1.85·29-s − 1.07·31-s − 0.353·32-s + 1.39·33-s − 0.685·34-s − 2/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30625\)    =    \(5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.95267\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9720920707\)
\(L(\frac12)\) \(\approx\) \(0.9720920707\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_d
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_v
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_w
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_s
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$D_{4}$ \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_cf
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_ba
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_ew
43$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_dt
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_du
61$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_di
67$D_{4}$ \( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_fd
71$D_{4}$ \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_dx
73$D_{4}$ \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.73.aw_kc
79$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.79.a_bh
83$D_{4}$ \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_es
89$D_{4}$ \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \) 2.89.abe_pi
97$D_{4}$ \( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_hq
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.01110474154463579187197850579, −12.36216294739086119519615816013, −12.28420944512452191397895646469, −11.53100478515449172831423995482, −10.62542567245760530310988346609, −10.29640987970285833536666456832, −9.615821546120246046133646690672, −9.498905136339927425339439551313, −8.841567132457867444396120965366, −8.643314798444274129168571783166, −7.931270676336452752173931234973, −7.72279593523330815772616751509, −6.50672188460512835567491340138, −6.48470752154622427729117035079, −5.39509543543337615067873355224, −4.59952085184925284287560304256, −3.69334306737398303061109220387, −3.64739169163803135623810162905, −2.43635580827897145919702985945, −1.09895466840553961393358200917, 1.09895466840553961393358200917, 2.43635580827897145919702985945, 3.64739169163803135623810162905, 3.69334306737398303061109220387, 4.59952085184925284287560304256, 5.39509543543337615067873355224, 6.48470752154622427729117035079, 6.50672188460512835567491340138, 7.72279593523330815772616751509, 7.931270676336452752173931234973, 8.643314798444274129168571783166, 8.841567132457867444396120965366, 9.498905136339927425339439551313, 9.615821546120246046133646690672, 10.29640987970285833536666456832, 10.62542567245760530310988346609, 11.53100478515449172831423995482, 12.28420944512452191397895646469, 12.36216294739086119519615816013, 13.01110474154463579187197850579

Graph of the $Z$-function along the critical line