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Properties

Label 2-1008-63.58-c1-0-38
Degree $2$
Conductor $1008$
Sign $0.858 + 0.513i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.46 − 0.918i)3-s + (1.89 + 3.29i)5-s + (0.841 − 2.50i)7-s + (1.31 − 2.69i)9-s + (2.25 − 3.90i)11-s + (0.588 − 1.01i)13-s + (5.81 + 3.08i)15-s + (−2.95 − 5.12i)17-s + (−2.55 + 4.42i)19-s + (−1.06 − 4.45i)21-s + (−2.09 − 3.62i)23-s + (−4.71 + 8.17i)25-s + (−0.545 − 5.16i)27-s + (2.11 + 3.65i)29-s + 6.24·31-s + ⋯
L(s)  = 1  + (0.847 − 0.530i)3-s + (0.849 + 1.47i)5-s + (0.318 − 0.948i)7-s + (0.438 − 0.898i)9-s + (0.680 − 1.17i)11-s + (0.163 − 0.282i)13-s + (1.50 + 0.797i)15-s + (−0.717 − 1.24i)17-s + (−0.586 + 1.01i)19-s + (−0.232 − 0.972i)21-s + (−0.435 − 0.755i)23-s + (−0.943 + 1.63i)25-s + (−0.105 − 0.994i)27-s + (0.392 + 0.679i)29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.858 + 0.513i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606292795\)
\(L(\frac12)\) \(\approx\) \(2.606292795\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 + 0.918i)T \)
7 \( 1 + (-0.841 + 2.50i)T \)
good5 \( 1 + (-1.89 - 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.25 + 3.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.588 + 1.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.55 - 4.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.11 - 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.754 + 1.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 8.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + (-6.49 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 - 6.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + (-0.307 - 0.532i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.25 - 2.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 - 4.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00259436646958932643798320724, −9.023199212060538312730508390447, −8.176072918979678521756176275239, −7.29224018935741388031822573822, −6.53923069454374670163184226483, −6.07660139781291638444769282681, −4.34602411541564461266907260613, −3.27123027469508500963740613570, −2.58609914249893245366064242149, −1.24086133015837433458945298334, 1.74911703426590902572821792195, 2.25111712316668533543591173282, 4.09222202724474255718031294325, 4.64343325450669177939199389499, 5.53548972794523694872569758698, 6.54437850017555678591753086309, 7.87981696211343057046974035455, 8.829748053485329133576273935009, 8.969665636893458333448846346957, 9.731890006554915388091856111231

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