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Properties

Label 2-315-105.2-c1-0-3
Degree $2$
Conductor $315$
Sign $0.650 + 0.759i$
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  + (−0.695 − 2.59i)2-s + (−4.52 + 2.61i)4-s + (1.27 + 1.83i)5-s + (−1.50 + 2.17i)7-s + (6.14 + 6.14i)8-s + (3.88 − 4.58i)10-s + (3.60 − 2.07i)11-s + (−0.702 + 0.702i)13-s + (6.69 + 2.40i)14-s + (6.44 − 11.1i)16-s + (1.31 + 0.352i)17-s + (5.22 + 3.01i)19-s + (−10.5 − 5.00i)20-s + (−7.90 − 7.90i)22-s + (0.205 − 0.0550i)23-s + ⋯
L(s)  = 1  + (−0.492 − 1.83i)2-s + (−2.26 + 1.30i)4-s + (0.569 + 0.822i)5-s + (−0.569 + 0.821i)7-s + (2.17 + 2.17i)8-s + (1.23 − 1.44i)10-s + (1.08 − 0.626i)11-s + (−0.194 + 0.194i)13-s + (1.78 + 0.641i)14-s + (1.61 − 2.79i)16-s + (0.319 + 0.0855i)17-s + (1.19 + 0.692i)19-s + (−2.36 − 1.11i)20-s + (−1.68 − 1.68i)22-s + (0.0428 − 0.0114i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.852045 - 0.392155i\)
\(L(\frac12)\) \(\approx\) \(0.852045 - 0.392155i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.83i)T \)
7 \( 1 + (1.50 - 2.17i)T \)
good2 \( 1 + (0.695 + 2.59i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-3.60 + 2.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.702 - 0.702i)T - 13iT^{2} \)
17 \( 1 + (-1.31 - 0.352i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.205 + 0.0550i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.879T + 29T^{2} \)
31 \( 1 + (-4.76 - 8.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.13 - 0.840i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.25iT - 41T^{2} \)
43 \( 1 + (4.28 - 4.28i)T - 43iT^{2} \)
47 \( 1 + (-0.0279 - 0.104i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.29 + 4.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.113 + 0.195i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.96 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.825 + 3.07i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.78iT - 71T^{2} \)
73 \( 1 + (6.60 + 1.77i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.88 - 1.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.75 - 3.75i)T + 83iT^{2} \)
89 \( 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.88 - 3.88i)T + 97iT^{2} \)
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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

−11.64925182294519774996742178653, −10.56397718179861222124608980444, −9.777532495478175120781824855725, −9.219432240801524212603631958176, −8.237462818600208122361085952590, −6.66405042596414151483771232001, −5.33398666731274032329372758380, −3.59991498204569265807009744707, −2.93374556248231806305337119372, −1.56560471352219215960567494840, 0.934433974218193797945875637973, 4.10913912423200682631138322572, 5.08632895095060095937911089290, 6.11433201013670993709758497956, 6.99311247266673212468857141002, 7.76537417741093139604073121272, 8.956688489385521609811856772929, 9.570655322053557091643685044327, 10.16914540084063677208953730609, 11.97720194985623203652851293476

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