Properties

Label 4-704e2-1.1-c1e2-0-4
Degree $4$
Conductor $495616$
Sign $1$
Analytic cond. $31.6009$
Root an. cond. $2.37096$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s + 7·9-s + 4·11-s − 8·15-s + 8·23-s − 3·25-s + 4·27-s + 16·33-s + 6·37-s − 14·45-s + 6·49-s − 4·53-s − 8·55-s + 12·59-s + 4·67-s + 32·69-s + 24·71-s − 12·75-s − 8·81-s − 26·89-s + 6·97-s + 28·99-s − 16·103-s + 24·111-s − 10·113-s − 16·115-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 7/3·9-s + 1.20·11-s − 2.06·15-s + 1.66·23-s − 3/5·25-s + 0.769·27-s + 2.78·33-s + 0.986·37-s − 2.08·45-s + 6/7·49-s − 0.549·53-s − 1.07·55-s + 1.56·59-s + 0.488·67-s + 3.85·69-s + 2.84·71-s − 1.38·75-s − 8/9·81-s − 2.75·89-s + 0.609·97-s + 2.81·99-s − 1.57·103-s + 2.27·111-s − 0.940·113-s − 1.49·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(495616\)    =    \(2^{12} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(31.6009\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 495616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.040342453\)
\(L(\frac12)\) \(\approx\) \(4.040342453\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) 2.3.ae_j
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.c_h
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.23.ai_cb
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.ag_bv
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.43.a_cc
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.59.am_fx
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.ae_dl
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) 2.71.ay_kr
73$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.73.a_co
79$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.79.a_aek
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.89.ba_mt
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.97.ag_hv
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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

−8.527834757835193726747029498818, −8.130905320346842230293799253614, −7.80880047675026696994038925633, −7.38343675003183133018650001814, −6.75675677658737613905670147750, −6.62082714324451009339742326913, −5.66163434916591204950710996791, −5.16707365592353737946128191517, −4.30971861340138374132199477742, −3.95501966037150973513946344449, −3.64483685966352076698862940684, −3.01131321872163071924120162853, −2.60939963678351254594850012675, −1.92256304918733355678144886727, −0.989457762310743048355362377137, 0.989457762310743048355362377137, 1.92256304918733355678144886727, 2.60939963678351254594850012675, 3.01131321872163071924120162853, 3.64483685966352076698862940684, 3.95501966037150973513946344449, 4.30971861340138374132199477742, 5.16707365592353737946128191517, 5.66163434916591204950710996791, 6.62082714324451009339742326913, 6.75675677658737613905670147750, 7.38343675003183133018650001814, 7.80880047675026696994038925633, 8.130905320346842230293799253614, 8.527834757835193726747029498818

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