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Properties

Label 2-315-315.202-c1-0-20
Degree $2$
Conductor $315$
Sign $0.142 - 0.989i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−0.377 + 1.41i)2-s + (1.62 − 0.591i)3-s + (−0.113 − 0.0657i)4-s + (2.18 + 0.477i)5-s + (0.218 + 2.51i)6-s + (−0.794 + 2.52i)7-s + (−1.92 + 1.92i)8-s + (2.30 − 1.92i)9-s + (−1.49 + 2.89i)10-s + (−1.95 − 3.38i)11-s + (−0.224 − 0.0397i)12-s + (−0.354 + 0.0948i)13-s + (−3.25 − 2.07i)14-s + (3.83 − 0.513i)15-s + (−2.12 − 3.67i)16-s + (−1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (−0.267 + 0.997i)2-s + (0.939 − 0.341i)3-s + (−0.0569 − 0.0328i)4-s + (0.976 + 0.213i)5-s + (0.0892 + 1.02i)6-s + (−0.300 + 0.953i)7-s + (−0.681 + 0.681i)8-s + (0.766 − 0.641i)9-s + (−0.474 + 0.917i)10-s + (−0.589 − 1.02i)11-s + (−0.0647 − 0.0114i)12-s + (−0.0981 + 0.0263i)13-s + (−0.870 − 0.554i)14-s + (0.991 − 0.132i)15-s + (−0.530 − 0.919i)16-s + (−0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.142 - 0.989i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34061 + 1.16181i\)
\(L(\frac12)\) \(\approx\) \(1.34061 + 1.16181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 + 0.591i)T \)
5 \( 1 + (-2.18 - 0.477i)T \)
7 \( 1 + (0.794 - 2.52i)T \)
good2 \( 1 + (0.377 - 1.41i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (1.95 + 3.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.354 - 0.0948i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
23 \( 1 + (1.67 + 6.23i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.97 - 2.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.80 - 1.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0303 + 0.0303i)T + 37iT^{2} \)
41 \( 1 + (9.89 + 5.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.456 - 0.122i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.3 - 3.03i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.71 - 8.71i)T - 53iT^{2} \)
59 \( 1 + (-4.90 + 8.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.75 + 3.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.81 - 1.02i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + (-8.94 - 8.94i)T + 73iT^{2} \)
79 \( 1 + (3.15 - 1.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.53 + 13.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.00T + 89T^{2} \)
97 \( 1 + (11.7 + 3.15i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.10079242260506063563996934815, −10.77070169416995970295749191670, −9.569829758810521558572074403646, −8.784613961244852705878714444743, −8.196017227377614666570439508817, −7.01091383620141976693694327748, −6.19494180333511936615574520177, −5.36736911057600287931940769747, −3.13229859226065058316865098301, −2.26617427736801830711761909667, 1.58225891911505191696905473464, 2.66357461603409505885551582630, 3.84925897727628962872540616581, 5.19610599344519541575336104154, 6.78686912245315314156388876010, 7.70304591324743757625818613089, 9.153750140254963267878574022820, 9.913033308424520950448738903313, 10.08177145593587723865602055517, 11.20992492004363029639374000009

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