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Properties

Label 2-310-155.149-c1-0-3
Degree $2$
Conductor $310$
Sign $-0.883 - 0.467i$
Analytic cond. $2.47536$
Root an. cond. $1.57332$
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  + i·2-s + (0.307 − 0.177i)3-s − 4-s + (−2.17 + 0.536i)5-s + (0.177 + 0.307i)6-s + (0.754 − 0.435i)7-s − i·8-s + (−1.43 + 2.48i)9-s + (−0.536 − 2.17i)10-s + (−3.01 + 5.21i)11-s + (−0.307 + 0.177i)12-s + (2.42 + 1.40i)13-s + (0.435 + 0.754i)14-s + (−0.571 + 0.550i)15-s + 16-s + (−5.92 + 3.42i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.177 − 0.102i)3-s − 0.5·4-s + (−0.970 + 0.240i)5-s + (0.0724 + 0.125i)6-s + (0.285 − 0.164i)7-s − 0.353i·8-s + (−0.478 + 0.829i)9-s + (−0.169 − 0.686i)10-s + (−0.907 + 1.57i)11-s + (−0.0887 + 0.0512i)12-s + (0.673 + 0.388i)13-s + (0.116 + 0.201i)14-s + (−0.147 + 0.142i)15-s + 0.250·16-s + (−1.43 + 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $-0.883 - 0.467i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ -0.883 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188161 + 0.757564i\)
\(L(\frac12)\) \(\approx\) \(0.188161 + 0.757564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (2.17 - 0.536i)T \)
31 \( 1 + (-4.57 - 3.17i)T \)
good3 \( 1 + (-0.307 + 0.177i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.754 + 0.435i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.01 - 5.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.42 - 1.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.92 - 3.42i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.287 + 0.497i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.91iT - 23T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
37 \( 1 + (3.57 - 2.06i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.78 + 8.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 + 1.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.78iT - 47T^{2} \)
53 \( 1 + (-2.17 - 1.25i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0518 + 0.0898i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 1.95T + 61T^{2} \)
67 \( 1 + (-6.83 - 3.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.77 - 4.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.65 + 3.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.23 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.94 - 5.16i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + 16.4iT - 97T^{2} \)
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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

−12.23661210948576756596998308378, −10.93068683478783513579314027363, −10.43789413333662397390982980696, −8.834018635837893998329858345792, −8.182657387321109418957967516754, −7.34312626811378007122258066331, −6.48149414090256634729271603129, −4.88005336933158817082665813371, −4.22327331180118351089355269601, −2.38924855027887458357038824295, 0.55109990872336161460541034833, 2.87579296651444797094685320690, 3.72890832775107291061452890694, 5.03701148767281631341292298901, 6.22467260994892557973288352827, 7.84964525350217388210247472447, 8.555380256941685515253074310821, 9.256154904072437455098936707378, 10.70231508137902855262073798670, 11.38784767318076747286020713651

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