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Properties

Label 2-315-105.23-c1-0-11
Degree $2$
Conductor $315$
Sign $0.976 + 0.215i$
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  + (2.02 − 0.543i)2-s + (2.08 − 1.20i)4-s + (−0.530 + 2.17i)5-s + (2.41 − 1.07i)7-s + (0.609 − 0.609i)8-s + (0.105 + 4.69i)10-s + (4.56 − 2.63i)11-s + (−3.08 − 3.08i)13-s + (4.31 − 3.50i)14-s + (−1.50 + 2.60i)16-s + (−1.58 + 5.92i)17-s + (−0.715 − 0.413i)19-s + (1.51 + 5.17i)20-s + (7.82 − 7.82i)22-s + (−0.359 − 1.33i)23-s + ⋯
L(s)  = 1  + (1.43 − 0.384i)2-s + (1.04 − 0.602i)4-s + (−0.237 + 0.971i)5-s + (0.913 − 0.407i)7-s + (0.215 − 0.215i)8-s + (0.0332 + 1.48i)10-s + (1.37 − 0.794i)11-s + (−0.855 − 0.855i)13-s + (1.15 − 0.936i)14-s + (−0.376 + 0.651i)16-s + (−0.385 + 1.43i)17-s + (−0.164 − 0.0948i)19-s + (0.337 + 1.15i)20-s + (1.66 − 1.66i)22-s + (−0.0748 − 0.279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70252 - 0.295182i\)
\(L(\frac12)\) \(\approx\) \(2.70252 - 0.295182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.530 - 2.17i)T \)
7 \( 1 + (-2.41 + 1.07i)T \)
good2 \( 1 + (-2.02 + 0.543i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-4.56 + 2.63i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.08 + 3.08i)T + 13iT^{2} \)
17 \( 1 + (1.58 - 5.92i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.715 + 0.413i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.359 + 1.33i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.45T + 29T^{2} \)
31 \( 1 + (2.97 + 5.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.568 + 2.12i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.00iT - 41T^{2} \)
43 \( 1 + (1.73 + 1.73i)T + 43iT^{2} \)
47 \( 1 + (4.17 - 1.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.65 - 1.78i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.07 - 7.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.12 + 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.67 - 1.25i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.40iT - 71T^{2} \)
73 \( 1 + (2.80 - 10.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.72 - 4.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.36 - 2.36i)T - 83iT^{2} \)
89 \( 1 + (-4.69 + 8.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.63 + 5.63i)T - 97iT^{2} \)
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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

−11.53453417283482039893038654857, −11.15028639619375670311363309304, −10.25891307905600896833742273147, −8.701169081801625817864742579826, −7.57580714096824369577213068439, −6.43216239312135039360168086667, −5.56816043485964726462170012948, −4.18389373610794297063821381604, −3.55958197883868474372353285563, −2.10420417983479401025210734696, 1.95863721750920251611037641404, 3.87443511087448011720975189159, 4.75131085505416914891437153606, 5.28141529107532542911969525686, 6.71741321039394054771787322891, 7.50095075727738489502295879939, 8.989861276449559655539519722924, 9.506972643628470318395364954832, 11.58461025083228006889023419522, 11.78185349966316742525362395068

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