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Properties

Label 2-315-63.4-c1-0-20
Degree $2$
Conductor $315$
Sign $0.258 + 0.965i$
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  + (0.859 − 1.48i)2-s + (−1.31 + 1.13i)3-s + (−0.477 − 0.827i)4-s − 5-s + (0.557 + 2.92i)6-s + (0.594 − 2.57i)7-s + 1.79·8-s + (0.438 − 2.96i)9-s + (−0.859 + 1.48i)10-s + 4.60·11-s + (1.56 + 0.544i)12-s + (0.944 − 1.63i)13-s + (−3.32 − 3.10i)14-s + (1.31 − 1.13i)15-s + (2.49 − 4.32i)16-s + (0.371 − 0.643i)17-s + ⋯
L(s)  = 1  + (0.607 − 1.05i)2-s + (−0.757 + 0.653i)3-s + (−0.238 − 0.413i)4-s − 0.447·5-s + (0.227 + 1.19i)6-s + (0.224 − 0.974i)7-s + 0.634·8-s + (0.146 − 0.989i)9-s + (−0.271 + 0.470i)10-s + 1.38·11-s + (0.451 + 0.157i)12-s + (0.262 − 0.453i)13-s + (−0.889 − 0.828i)14-s + (0.338 − 0.292i)15-s + (0.624 − 1.08i)16-s + (0.0901 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19509 - 0.917249i\)
\(L(\frac12)\) \(\approx\) \(1.19509 - 0.917249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.31 - 1.13i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.594 + 2.57i)T \)
good2 \( 1 + (-0.859 + 1.48i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 + (-0.944 + 1.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.643i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.518 + 0.897i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.77T + 23T^{2} \)
29 \( 1 + (-4.71 - 8.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.589 + 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 + 4.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.72 - 6.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.26 + 2.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.959 + 1.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.94 - 6.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.55 - 4.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 - 3.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.72 - 6.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.30T + 71T^{2} \)
73 \( 1 + (-3.33 + 5.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.60 + 4.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.45 + 14.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.48 - 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.80 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.58662499296333682755568161009, −10.67522525402731082467407002742, −10.18345587937111536932514845660, −8.925572869789081540573342028360, −7.48983184305466971340619743554, −6.44622948380915535846971906114, −4.95489132037016767519578822780, −4.08390755882758172087072921333, −3.41849008419016696654107753857, −1.19517108922563717105603753444, 1.75721730766785785179289079371, 4.06781540411763013613263535727, 5.11900870444854349006137435311, 6.22372577511234958616938167010, 6.57791256297153242606906153005, 7.80769330889550994765229075581, 8.589051714702343876217031303729, 10.04966025699997634945098370495, 11.38141926308841069250933709700, 11.88534244002061330796463690842

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