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Properties

Label 4-2366e2-1.1-c1e2-0-11
Degree $4$
Conductor $5597956$
Sign $1$
Analytic cond. $356.930$
Root an. cond. $4.34656$
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  + 6·3-s − 4-s + 21·9-s − 6·12-s + 16-s + 8·17-s − 10·23-s + 10·25-s + 54·27-s + 8·29-s − 21·36-s + 24·43-s + 6·48-s − 49-s + 48·51-s − 8·53-s + 26·61-s − 64-s − 8·68-s − 60·69-s + 60·75-s − 34·79-s + 108·81-s + 48·87-s + 10·92-s − 10·100-s − 30·101-s + ⋯
L(s)  = 1  + 3.46·3-s − 1/2·4-s + 7·9-s − 1.73·12-s + 1/4·16-s + 1.94·17-s − 2.08·23-s + 2·25-s + 10.3·27-s + 1.48·29-s − 7/2·36-s + 3.65·43-s + 0.866·48-s − 1/7·49-s + 6.72·51-s − 1.09·53-s + 3.32·61-s − 1/8·64-s − 0.970·68-s − 7.22·69-s + 6.92·75-s − 3.82·79-s + 12·81-s + 5.14·87-s + 1.04·92-s − 100-s − 2.98·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5597956\)    =    \(2^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(356.930\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5597956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.53717773\)
\(L(\frac12)\) \(\approx\) \(11.53717773\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + T^{2} \)
13 \( 1 \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.3.ag_p
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.23.k_ct
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.31.a_acj
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.41.a_ab
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.67.a_an
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \) 2.73.a_adt
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.79.bi_rf
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.89.a_s
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \) 2.97.a_agn
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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

−8.992166888515760021796512366338, −8.757429791841452776499381255534, −8.256982606396820526072971501411, −8.253872093571264453728959992081, −7.87588782054048980035670756949, −7.52522696087795979749684072830, −7.04928451668779594714168874877, −6.85482033274511253369803081237, −6.07473505083103519065892648071, −5.60005209743660177999034422076, −5.10751188393720479411464338502, −4.34209827872473544465693988349, −4.04636616902717464462048313865, −3.99619573043284449482670082987, −3.22920475486403225253778578650, −2.94345239625634532172811711379, −2.60010684508417576200628868167, −2.22944699826617285532981336199, −1.31072713296460928993573746125, −1.11408228987987545415741553574, 1.11408228987987545415741553574, 1.31072713296460928993573746125, 2.22944699826617285532981336199, 2.60010684508417576200628868167, 2.94345239625634532172811711379, 3.22920475486403225253778578650, 3.99619573043284449482670082987, 4.04636616902717464462048313865, 4.34209827872473544465693988349, 5.10751188393720479411464338502, 5.60005209743660177999034422076, 6.07473505083103519065892648071, 6.85482033274511253369803081237, 7.04928451668779594714168874877, 7.52522696087795979749684072830, 7.87588782054048980035670756949, 8.253872093571264453728959992081, 8.256982606396820526072971501411, 8.757429791841452776499381255534, 8.992166888515760021796512366338

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