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Properties

Label 2-2e2-4.3-c30-0-5
Degree $2$
Conductor $4$
Sign $-0.769 - 0.638i$
Analytic cond. $22.8057$
Root an. cond. $4.77553$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.08e4 + 1.11e4i)2-s + 1.23e7i·3-s + (8.26e8 + 6.85e8i)4-s + 1.48e9·5-s + (−1.36e11 + 3.79e11i)6-s + 2.60e12i·7-s + (1.78e13 + 3.03e13i)8-s + 5.42e13·9-s + (4.57e13 + 1.65e13i)10-s + 2.83e15i·11-s + (−8.44e15 + 1.01e16i)12-s − 4.46e16·13-s + (−2.89e16 + 8.02e16i)14-s + 1.82e16i·15-s + (2.13e17 + 1.13e18i)16-s − 8.36e17·17-s + ⋯
L(s)  = 1  + (0.940 + 0.339i)2-s + 0.858i·3-s + (0.769 + 0.638i)4-s + 0.0486·5-s + (−0.291 + 0.807i)6-s + 0.548i·7-s + (0.507 + 0.861i)8-s + 0.263·9-s + (0.0457 + 0.0165i)10-s + 0.677i·11-s + (−0.547 + 0.660i)12-s − 0.873·13-s + (−0.185 + 0.515i)14-s + 0.0417i·15-s + (0.185 + 0.982i)16-s − 0.292·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(22.8057\)
Root analytic conductor: \(4.77553\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :15),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(1.11045 + 3.07874i\)
\(L(\frac12)\) \(\approx\) \(1.11045 + 3.07874i\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.08e4 - 1.11e4i)T \)
good3 \( 1 - 1.23e7iT - 2.05e14T^{2} \)
5 \( 1 - 1.48e9T + 9.31e20T^{2} \)
7 \( 1 - 2.60e12iT - 2.25e25T^{2} \)
11 \( 1 - 2.83e15iT - 1.74e31T^{2} \)
13 \( 1 + 4.46e16T + 2.61e33T^{2} \)
17 \( 1 + 8.36e17T + 8.19e36T^{2} \)
19 \( 1 + 2.36e19iT - 2.30e38T^{2} \)
23 \( 1 - 4.21e20iT - 7.10e40T^{2} \)
29 \( 1 + 3.89e21T + 7.44e43T^{2} \)
31 \( 1 + 8.50e21iT - 5.50e44T^{2} \)
37 \( 1 - 5.15e23T + 1.11e47T^{2} \)
41 \( 1 - 1.29e24T + 2.41e48T^{2} \)
43 \( 1 + 5.30e23iT - 1.00e49T^{2} \)
47 \( 1 + 4.10e23iT - 1.45e50T^{2} \)
53 \( 1 - 1.20e26T + 5.34e51T^{2} \)
59 \( 1 + 7.03e26iT - 1.33e53T^{2} \)
61 \( 1 - 6.82e26T + 3.62e53T^{2} \)
67 \( 1 - 3.12e27iT - 6.05e54T^{2} \)
71 \( 1 - 1.56e27iT - 3.44e55T^{2} \)
73 \( 1 + 4.03e27T + 7.93e55T^{2} \)
79 \( 1 - 6.28e27iT - 8.48e56T^{2} \)
83 \( 1 - 6.96e28iT - 3.73e57T^{2} \)
89 \( 1 + 1.68e28T + 3.03e58T^{2} \)
97 \( 1 - 9.73e29T + 4.01e59T^{2} \)
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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

−17.43389248951044019461616434999, −15.75204988430589220757291591768, −14.97227463426912742636140439736, −13.10308589593013999339260090851, −11.47841937669131097058329356936, −9.552231214833322597501202527394, −7.27812742180458575124584889259, −5.33225321914222085204886330243, −4.12313648708514181878333792015, −2.35905368187672395096709788212, 0.825266930033462763756761978483, 2.27455263409269599240721214980, 4.14303515605218509319708656336, 6.08913629281141068756022892092, 7.52951711681911383615401829491, 10.29757914242653268455977691189, 12.05864613496654074058589163175, 13.24749043680544490804164423977, 14.52252462827608183187649710367, 16.48056333122222917352905819683

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