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Properties

Label 2-5-5.2-c22-0-5
Degree $2$
Conductor $5$
Sign $-0.974 - 0.224i$
Analytic cond. $15.3353$
Root an. cond. $3.91604$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.64e3 − 2.64e3i)2-s + (−7.56e4 + 7.56e4i)3-s + 9.76e6i·4-s + (3.47e7 + 3.43e7i)5-s + 3.99e8·6-s + (−1.89e9 − 1.89e9i)7-s + (1.47e10 − 1.47e10i)8-s + 1.99e10i·9-s + (−9.37e8 − 1.82e11i)10-s + 3.86e11·11-s + (−7.38e11 − 7.38e11i)12-s + (−8.29e11 + 8.29e11i)13-s + 1.00e13i·14-s + (−5.22e12 + 2.68e10i)15-s − 3.68e13·16-s + (−1.06e13 − 1.06e13i)17-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + (−0.427 + 0.427i)3-s + 2.32i·4-s + (0.710 + 0.703i)5-s + 1.10·6-s + (−0.958 − 0.958i)7-s + (1.71 − 1.71i)8-s + 0.635i·9-s + (−0.00937 − 1.82i)10-s + 1.35·11-s + (−0.994 − 0.994i)12-s + (−0.462 + 0.462i)13-s + 2.47i·14-s + (−0.603 + 0.00310i)15-s − 2.09·16-s + (−0.310 − 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.974 - 0.224i$
Analytic conductor: \(15.3353\)
Root analytic conductor: \(3.91604\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :11),\ -0.974 - 0.224i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(0.0142697 + 0.125358i\)
\(L(\frac12)\) \(\approx\) \(0.0142697 + 0.125358i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.47e7 - 3.43e7i)T \)
good2 \( 1 + (2.64e3 + 2.64e3i)T + 4.19e6iT^{2} \)
3 \( 1 + (7.56e4 - 7.56e4i)T - 3.13e10iT^{2} \)
7 \( 1 + (1.89e9 + 1.89e9i)T + 3.90e18iT^{2} \)
11 \( 1 - 3.86e11T + 8.14e22T^{2} \)
13 \( 1 + (8.29e11 - 8.29e11i)T - 3.21e24iT^{2} \)
17 \( 1 + (1.06e13 + 1.06e13i)T + 1.17e27iT^{2} \)
19 \( 1 + 5.02e13iT - 1.35e28T^{2} \)
23 \( 1 + (5.55e14 - 5.55e14i)T - 9.07e29iT^{2} \)
29 \( 1 + 8.97e15iT - 1.48e32T^{2} \)
31 \( 1 + 2.58e16T + 6.45e32T^{2} \)
37 \( 1 + (1.10e16 + 1.10e16i)T + 3.16e34iT^{2} \)
41 \( 1 + 1.07e18T + 3.02e35T^{2} \)
43 \( 1 + (-9.67e17 + 9.67e17i)T - 8.63e35iT^{2} \)
47 \( 1 + (2.25e18 + 2.25e18i)T + 6.11e36iT^{2} \)
53 \( 1 + (-2.87e18 + 2.87e18i)T - 8.59e37iT^{2} \)
59 \( 1 + 2.44e19iT - 9.09e38T^{2} \)
61 \( 1 - 1.19e19T + 1.89e39T^{2} \)
67 \( 1 + (4.25e19 + 4.25e19i)T + 1.49e40iT^{2} \)
71 \( 1 + 2.43e19T + 5.34e40T^{2} \)
73 \( 1 + (3.97e18 - 3.97e18i)T - 9.84e40iT^{2} \)
79 \( 1 + 8.70e20iT - 5.59e41T^{2} \)
83 \( 1 + (4.58e20 - 4.58e20i)T - 1.65e42iT^{2} \)
89 \( 1 + 1.81e20iT - 7.70e42T^{2} \)
97 \( 1 + (-1.09e21 - 1.09e21i)T + 5.11e43iT^{2} \)
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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

−17.41000914095517085966927859342, −16.61816424110934450390214016388, −13.56989328777110947478129595174, −11.54417806156236448640187786783, −10.30697154530131966450664296402, −9.438820972141123391684120994204, −6.99443134218295240696445756470, −3.68043741136163766452912237273, −1.92712319062070680625665993864, −0.085527311671460631049339786635, 1.37122635440157245906380541855, 5.77943794605001315779160989977, 6.61080726809533948396533366176, 8.801037446995013202629117785065, 9.694728686301462061683619411263, 12.43698843406378862300077121376, 14.74996211603048841628183543463, 16.31411642142737793271114138635, 17.34797471616383695304527808315, 18.40421767388550151183443218151

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