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Properties

Label 4-3510e2-1.1-c1e2-0-1
Degree $4$
Conductor $12320100$
Sign $1$
Analytic cond. $785.540$
Root an. cond. $5.29409$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s − 2·7-s − 8-s − 10-s + 11-s + 13-s − 2·14-s − 16-s − 10·17-s + 6·19-s + 22-s + 26-s + 8·29-s − 6·31-s − 10·34-s + 2·35-s − 16·37-s + 6·38-s + 40-s + 7·41-s − 5·43-s − 12·47-s + 7·49-s − 4·53-s − 55-s + 2·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s − 1/4·16-s − 2.42·17-s + 1.37·19-s + 0.213·22-s + 0.196·26-s + 1.48·29-s − 1.07·31-s − 1.71·34-s + 0.338·35-s − 2.63·37-s + 0.973·38-s + 0.158·40-s + 1.09·41-s − 0.762·43-s − 1.75·47-s + 49-s − 0.549·53-s − 0.134·55-s + 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12320100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12320100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12320100\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(785.540\)
Root analytic conductor: \(5.29409\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12320100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4363033361\)
\(L(\frac12)\) \(\approx\) \(0.4363033361\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bj
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_f
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.41.ah_i
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.f_as
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_abi
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.67.ap_gc
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−8.836378011511291963881294864645, −8.431420282665371087248460688158, −8.212886836932968270747120215996, −7.29595125668086425625171373231, −7.22345899988281511316687118548, −7.01677364232228273208716711719, −6.38208229646195204086821196752, −6.24563690617456204549495509686, −5.77545485277356540238842344107, −5.31925618873662863103241807560, −4.71347161161108946780578017702, −4.68573705182230860107779556324, −4.17996937250630235286214880096, −3.66493443569365931540491263894, −3.18883287848661479335573067989, −3.12908415246417627941448169128, −2.38271564793668515133924981672, −1.81689252393259546455818039967, −1.23953861718959448476971005089, −0.17790834389363177406800796806, 0.17790834389363177406800796806, 1.23953861718959448476971005089, 1.81689252393259546455818039967, 2.38271564793668515133924981672, 3.12908415246417627941448169128, 3.18883287848661479335573067989, 3.66493443569365931540491263894, 4.17996937250630235286214880096, 4.68573705182230860107779556324, 4.71347161161108946780578017702, 5.31925618873662863103241807560, 5.77545485277356540238842344107, 6.24563690617456204549495509686, 6.38208229646195204086821196752, 7.01677364232228273208716711719, 7.22345899988281511316687118548, 7.29595125668086425625171373231, 8.212886836932968270747120215996, 8.431420282665371087248460688158, 8.836378011511291963881294864645

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