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Properties

Label 2-5-5.2-c26-0-7
Degree $2$
Conductor $5$
Sign $0.963 - 0.269i$
Analytic cond. $21.4146$
Root an. cond. $4.62759$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.34e3 + 6.34e3i)2-s + (1.05e5 − 1.05e5i)3-s + 1.34e7i·4-s + (−1.17e9 − 3.38e8i)5-s + 1.33e9·6-s + (1.02e10 + 1.02e10i)7-s + (3.40e11 − 3.40e11i)8-s + 2.51e12i·9-s + (−5.29e12 − 9.59e12i)10-s + 5.16e13·11-s + (1.41e12 + 1.41e12i)12-s + (3.44e14 − 3.44e14i)13-s + 1.29e14i·14-s + (−1.58e14 + 8.77e13i)15-s + 5.22e15·16-s + (−3.40e15 − 3.40e15i)17-s + ⋯
L(s)  = 1  + (0.774 + 0.774i)2-s + (0.0659 − 0.0659i)3-s + 0.200i·4-s + (−0.960 − 0.277i)5-s + 0.102·6-s + (0.105 + 0.105i)7-s + (0.619 − 0.619i)8-s + 0.991i·9-s + (−0.529 − 0.959i)10-s + 1.49·11-s + (0.0132 + 0.0132i)12-s + (1.13 − 1.13i)13-s + 0.163i·14-s + (−0.0816 + 0.0450i)15-s + 1.16·16-s + (−0.344 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.963 - 0.269i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ 0.963 - 0.269i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(2.93138 + 0.401891i\)
\(L(\frac12)\) \(\approx\) \(2.93138 + 0.401891i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.17e9 + 3.38e8i)T \)
good2 \( 1 + (-6.34e3 - 6.34e3i)T + 6.71e7iT^{2} \)
3 \( 1 + (-1.05e5 + 1.05e5i)T - 2.54e12iT^{2} \)
7 \( 1 + (-1.02e10 - 1.02e10i)T + 9.38e21iT^{2} \)
11 \( 1 - 5.16e13T + 1.19e27T^{2} \)
13 \( 1 + (-3.44e14 + 3.44e14i)T - 9.17e28iT^{2} \)
17 \( 1 + (3.40e15 + 3.40e15i)T + 9.81e31iT^{2} \)
19 \( 1 - 1.68e16iT - 1.76e33T^{2} \)
23 \( 1 + (-5.57e17 + 5.57e17i)T - 2.54e35iT^{2} \)
29 \( 1 - 8.40e18iT - 1.05e38T^{2} \)
31 \( 1 - 1.77e18T + 5.96e38T^{2} \)
37 \( 1 + (1.98e20 + 1.98e20i)T + 5.93e40iT^{2} \)
41 \( 1 + 9.08e20T + 8.55e41T^{2} \)
43 \( 1 + (2.42e20 - 2.42e20i)T - 2.95e42iT^{2} \)
47 \( 1 + (-4.53e21 - 4.53e21i)T + 2.98e43iT^{2} \)
53 \( 1 + (3.70e20 - 3.70e20i)T - 6.77e44iT^{2} \)
59 \( 1 + 1.54e23iT - 1.10e46T^{2} \)
61 \( 1 - 5.28e22T + 2.62e46T^{2} \)
67 \( 1 + (-4.58e23 - 4.58e23i)T + 3.00e47iT^{2} \)
71 \( 1 - 1.77e24T + 1.35e48T^{2} \)
73 \( 1 + (5.16e23 - 5.16e23i)T - 2.79e48iT^{2} \)
79 \( 1 - 3.36e24iT - 2.17e49T^{2} \)
83 \( 1 + (3.58e24 - 3.58e24i)T - 7.87e49iT^{2} \)
89 \( 1 - 8.14e24iT - 4.83e50T^{2} \)
97 \( 1 + (1.12e25 + 1.12e25i)T + 4.52e51iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.67288039568125903974945990165, −15.56950208811393955639812978683, −14.27201279077979685154556016092, −12.78571578793878889219884662096, −10.92465778075415183042086274347, −8.435749797022008171764370309263, −6.85581088872235281237419008457, −5.13013130693336053708646156598, −3.73874308008194130560306803700, −1.04870985397813618276553893687, 1.31777263456690335581655679494, 3.45582130573112989453117178879, 4.18293972763044215624186392517, 6.75465658834916545104056585874, 8.842752100780797646334327944795, 11.28074465160166684557379872480, 11.97941775421885451683251189063, 13.77140444582512550035974513639, 15.20186947850326652475811946999, 17.08663008752086294976087826858

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