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Properties

Label 2-315-45.34-c1-0-14
Degree $2$
Conductor $315$
Sign $0.589 + 0.807i$
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  + (−1.03 − 0.596i)2-s + (−0.463 + 1.66i)3-s + (−0.288 − 0.500i)4-s + (1.52 − 1.63i)5-s + (1.47 − 1.44i)6-s + (0.866 + 0.5i)7-s + 3.07i·8-s + (−2.57 − 1.54i)9-s + (−2.54 + 0.785i)10-s + (2.02 − 3.50i)11-s + (0.968 − 0.250i)12-s + (−2.53 + 1.46i)13-s + (−0.596 − 1.03i)14-s + (2.03 + 3.29i)15-s + (1.25 − 2.17i)16-s − 3.25i·17-s + ⋯
L(s)  = 1  + (−0.730 − 0.421i)2-s + (−0.267 + 0.963i)3-s + (−0.144 − 0.250i)4-s + (0.680 − 0.732i)5-s + (0.601 − 0.590i)6-s + (0.327 + 0.188i)7-s + 1.08i·8-s + (−0.857 − 0.515i)9-s + (−0.805 + 0.248i)10-s + (0.609 − 1.05i)11-s + (0.279 − 0.0723i)12-s + (−0.703 + 0.405i)13-s + (−0.159 − 0.276i)14-s + (0.524 + 0.851i)15-s + (0.313 − 0.543i)16-s − 0.790i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.589 + 0.807i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.589 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761397 - 0.386948i\)
\(L(\frac12)\) \(\approx\) \(0.761397 - 0.386948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.463 - 1.66i)T \)
5 \( 1 + (-1.52 + 1.63i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (1.03 + 0.596i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.02 + 3.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.53 - 1.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.25iT - 17T^{2} \)
19 \( 1 - 3.90T + 19T^{2} \)
23 \( 1 + (-7.50 + 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.27 + 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.72 - 2.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.39iT - 37T^{2} \)
41 \( 1 + (-3.10 - 5.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.49 + 3.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.71 + 5.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.38iT - 53T^{2} \)
59 \( 1 + (-0.624 - 1.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 4.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.47 - 4.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.24T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (-2.12 + 3.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.5 - 6.09i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.60T + 89T^{2} \)
97 \( 1 + (-1.37 - 0.791i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.46224330824250913194039468068, −10.35227375167569985314942852360, −9.629265377774138306345833398566, −8.991662309448473643475853614876, −8.338820053683991427824692167919, −6.40925188726255461610794866276, −5.24510038963226652386610753795, −4.71559635578713521380706875527, −2.80075285385301283737338721828, −0.933081232700070085401096209161, 1.51510758189844259652524126042, 3.15784786221627090748482941777, 5.01013316498864941944280574786, 6.35249717771380817496194704582, 7.22553387114738107035884430251, 7.63097081616547648717210936241, 8.944424345262192641390768926361, 9.779452725487598544789190843920, 10.75849445714525456515691941007, 11.86633694831952091357159197221

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