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Properties

Label 2-315-63.47-c1-0-17
Degree $2$
Conductor $315$
Sign $0.980 + 0.197i$
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  + (0.441 − 0.254i)2-s + (1.05 − 1.37i)3-s + (−0.870 + 1.50i)4-s + 5-s + (0.115 − 0.875i)6-s + (2.05 + 1.66i)7-s + 1.90i·8-s + (−0.775 − 2.89i)9-s + (0.441 − 0.254i)10-s + 4.00i·11-s + (1.15 + 2.78i)12-s + (5.51 − 3.18i)13-s + (1.33 + 0.213i)14-s + (1.05 − 1.37i)15-s + (−1.25 − 2.17i)16-s + (−1.73 − 3.01i)17-s + ⋯
L(s)  = 1  + (0.312 − 0.180i)2-s + (0.608 − 0.793i)3-s + (−0.435 + 0.753i)4-s + 0.447·5-s + (0.0471 − 0.357i)6-s + (0.776 + 0.630i)7-s + 0.674i·8-s + (−0.258 − 0.966i)9-s + (0.139 − 0.0806i)10-s + 1.20i·11-s + (0.332 + 0.803i)12-s + (1.53 − 0.883i)13-s + (0.355 + 0.0569i)14-s + (0.272 − 0.354i)15-s + (−0.313 − 0.542i)16-s + (−0.421 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89311 - 0.188482i\)
\(L(\frac12)\) \(\approx\) \(1.89311 - 0.188482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.05 + 1.37i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.05 - 1.66i)T \)
good2 \( 1 + (-0.441 + 0.254i)T + (1 - 1.73i)T^{2} \)
11 \( 1 - 4.00iT - 11T^{2} \)
13 \( 1 + (-5.51 + 3.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.73 + 3.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.99 + 2.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.03iT - 23T^{2} \)
29 \( 1 + (4.74 + 2.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.66 - 2.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.92 - 3.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.26 + 5.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.188 - 0.325i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0931 - 0.161i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.99 - 2.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 - 9.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.03 + 4.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.98 + 5.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 + (10.7 - 6.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.816 - 1.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.417 - 0.722i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.95 - 6.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.8 + 8.54i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.91358046435566752450258284673, −10.96415563975412262701914461785, −9.441689883122529711239965746806, −8.644580405392799449639309758420, −7.993179044826623463530316076707, −6.93093908959369250612630065595, −5.62465227207063092901902676313, −4.39370505771987870604369018065, −3.00707556753238017905810047210, −1.87210694102997994117085903919, 1.64995815259812350083374506117, 3.73423682631893087499120500053, 4.42775097447129586527627999792, 5.66951779440249443289732728052, 6.51854437779855002922452550967, 8.370118634904460557510484168359, 8.711890175364714629988050315767, 9.876442553382686097029335421278, 10.83475208589108397099680689973, 11.12808229628335016791947581161

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