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Properties

Label 2-2e2-4.3-c38-0-17
Degree $2$
Conductor $4$
Sign $-0.957 + 0.289i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.18e5 − 7.66e4i)2-s − 2.28e9i·3-s + (2.63e11 − 7.95e10i)4-s + 1.45e13·5-s + (−1.75e14 − 1.18e15i)6-s − 8.90e15i·7-s + (1.30e17 − 6.14e16i)8-s − 3.85e18·9-s + (7.56e18 − 1.11e18i)10-s − 4.38e19i·11-s + (−1.81e20 − 6.00e20i)12-s + 1.87e21·13-s + (−6.82e20 − 4.61e21i)14-s − 3.33e22i·15-s + (6.29e22 − 4.18e22i)16-s + 4.37e22·17-s + ⋯
L(s)  = 1  + (0.989 − 0.146i)2-s − 1.96i·3-s + (0.957 − 0.289i)4-s + 0.765·5-s + (−0.287 − 1.94i)6-s − 0.781i·7-s + (0.904 − 0.426i)8-s − 2.85·9-s + (0.756 − 0.111i)10-s − 0.716i·11-s + (−0.568 − 1.87i)12-s + 1.28·13-s + (−0.114 − 0.773i)14-s − 1.50i·15-s + (0.832 − 0.553i)16-s + 0.182·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(4.431670945\)
\(L(\frac12)\) \(\approx\) \(4.431670945\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.18e5 + 7.66e4i)T \)
good3 \( 1 + 2.28e9iT - 1.35e18T^{2} \)
5 \( 1 - 1.45e13T + 3.63e26T^{2} \)
7 \( 1 + 8.90e15iT - 1.29e32T^{2} \)
11 \( 1 + 4.38e19iT - 3.74e39T^{2} \)
13 \( 1 - 1.87e21T + 2.13e42T^{2} \)
17 \( 1 - 4.37e22T + 5.71e46T^{2} \)
19 \( 1 - 2.11e24iT - 3.91e48T^{2} \)
23 \( 1 + 2.46e25iT - 5.56e51T^{2} \)
29 \( 1 - 3.71e27T + 3.72e55T^{2} \)
31 \( 1 - 1.46e28iT - 4.69e56T^{2} \)
37 \( 1 + 3.14e29T + 3.90e59T^{2} \)
41 \( 1 + 1.59e30T + 1.93e61T^{2} \)
43 \( 1 - 1.13e31iT - 1.17e62T^{2} \)
47 \( 1 - 3.60e31iT - 3.46e63T^{2} \)
53 \( 1 - 1.01e32T + 3.33e65T^{2} \)
59 \( 1 + 1.33e33iT - 1.96e67T^{2} \)
61 \( 1 + 7.35e33T + 6.95e67T^{2} \)
67 \( 1 + 3.46e34iT - 2.45e69T^{2} \)
71 \( 1 + 2.59e35iT - 2.22e70T^{2} \)
73 \( 1 - 3.29e35T + 6.40e70T^{2} \)
79 \( 1 - 9.78e35iT - 1.28e72T^{2} \)
83 \( 1 - 3.96e36iT - 8.41e72T^{2} \)
89 \( 1 - 7.10e36T + 1.19e74T^{2} \)
97 \( 1 + 6.10e37T + 3.14e75T^{2} \)
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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

−14.01658361018422937568803449258, −13.61804487556671420935223985635, −12.38206456419200671863230057659, −10.92790269072297201467733070916, −8.051517000891010760462529602711, −6.57787586758756629829930918268, −5.79878221342471916648404737871, −3.26283231233509826296445256764, −1.75982141615606736504321683882, −0.951830073367912053255610913156, 2.40757062861764049196904895768, 3.73941554332183683532677625878, 5.02742455769889833760208088245, 5.98251848690047825660799637813, 8.862825479601209855906040637652, 10.27171631226097375358988130523, 11.59137262605328788602846028551, 13.72951388581297315323903493862, 15.14184612621685860583662437105, 15.85179190524879051491514594316

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