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Properties

Label 2-310-31.26-c2-0-8
Degree $2$
Conductor $310$
Sign $0.987 - 0.156i$
Analytic cond. $8.44688$
Root an. cond. $2.90635$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (2.61 + 1.51i)3-s + 2.00·4-s + (1.11 + 1.93i)5-s + (−3.70 − 2.13i)6-s + (2.32 − 4.02i)7-s − 2.82·8-s + (0.0748 + 0.129i)9-s + (−1.58 − 2.73i)10-s + (8.60 − 4.96i)11-s + (5.23 + 3.02i)12-s + (15.0 − 8.70i)13-s + (−3.28 + 5.69i)14-s + 6.76i·15-s + 4.00·16-s + (1.41 + 0.816i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.873 + 0.504i)3-s + 0.500·4-s + (0.223 + 0.387i)5-s + (−0.617 − 0.356i)6-s + (0.332 − 0.575i)7-s − 0.353·8-s + (0.00831 + 0.0143i)9-s + (−0.158 − 0.273i)10-s + (0.782 − 0.451i)11-s + (0.436 + 0.252i)12-s + (1.15 − 0.669i)13-s + (−0.234 + 0.407i)14-s + 0.450i·15-s + 0.250·16-s + (0.0832 + 0.0480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(8.44688\)
Root analytic conductor: \(2.90635\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1),\ 0.987 - 0.156i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.80448 + 0.142375i\)
\(L(\frac12)\) \(\approx\) \(1.80448 + 0.142375i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 + (-1.11 - 1.93i)T \)
31 \( 1 + (3.74 - 30.7i)T \)
good3 \( 1 + (-2.61 - 1.51i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-2.32 + 4.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.60 + 4.96i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-15.0 + 8.70i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.41 - 0.816i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.00 + 8.66i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 25.2iT - 529T^{2} \)
29 \( 1 - 20.9iT - 841T^{2} \)
37 \( 1 + (37.7 + 21.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-2.17 - 3.77i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.9 - 19.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 40.8T + 2.20e3T^{2} \)
53 \( 1 + (-30.4 + 17.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (25.4 - 43.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + 18.6iT - 3.72e3T^{2} \)
67 \( 1 + (58.0 + 100. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (17.3 + 30.0i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-15.7 + 9.10i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-46.2 - 26.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (63.7 - 36.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 135. iT - 7.92e3T^{2} \)
97 \( 1 + 122.T + 9.40e3T^{2} \)
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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

−11.05958103005596483229300923354, −10.56155469587216935854718883262, −9.388858294404559073108218305881, −8.839308820813854770182771506239, −7.896690603889232883795627636583, −6.84066782089416654818210889497, −5.67796007985220863142660183982, −3.86419463479291992933677749945, −3.06568138597555764103967043181, −1.25904688603614767036902795119, 1.43523083727403704644009104807, 2.42670258256543603649225162294, 4.05124150521652029458400791619, 5.70343137570282511993276388576, 6.79554557022187244408921828601, 7.898461871419687144191273544773, 8.725756260026273598863061174218, 9.143660104916414775935750354444, 10.35109214192310631008587003782, 11.50963373000159836126619048221

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