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Properties

Label 2-315-35.12-c1-0-16
Degree $2$
Conductor $315$
Sign $0.528 + 0.848i$
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  + (1.86 − 0.5i)2-s + (1.5 − 0.866i)4-s + (0.133 − 2.23i)5-s + (0.866 − 2.5i)7-s + (−0.366 + 0.366i)8-s + (−0.866 − 4.23i)10-s + (0.366 + 0.633i)11-s + (2 + 2i)13-s + (0.366 − 5.09i)14-s + (−2.23 + 3.86i)16-s + (−1 − 0.267i)17-s + (1.36 − 2.36i)19-s + (−1.73 − 3.46i)20-s + (1 + i)22-s + (1.86 + 6.96i)23-s + ⋯
L(s)  = 1  + (1.31 − 0.353i)2-s + (0.750 − 0.433i)4-s + (0.0599 − 0.998i)5-s + (0.327 − 0.944i)7-s + (−0.129 + 0.129i)8-s + (−0.273 − 1.33i)10-s + (0.110 + 0.191i)11-s + (0.554 + 0.554i)13-s + (0.0978 − 1.36i)14-s + (−0.558 + 0.966i)16-s + (−0.242 − 0.0649i)17-s + (0.313 − 0.542i)19-s + (−0.387 − 0.774i)20-s + (0.213 + 0.213i)22-s + (0.389 + 1.45i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.528 + 0.848i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.528 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17884 - 1.20989i\)
\(L(\frac12)\) \(\approx\) \(2.17884 - 1.20989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + (-1.86 + 0.5i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (1 + 0.267i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.36 + 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 6.96i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-0.464 + 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 - 5.83i)T - 43iT^{2} \)
47 \( 1 + (0.169 + 0.633i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.83 + 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.09 - 1.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.33 + 4.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.303 - 1.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 + (-0.928 + 3.46i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.83 - 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 - 3.09i)T + 83iT^{2} \)
89 \( 1 + (-8.33 + 14.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.92 - 7.92i)T - 97iT^{2} \)
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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

−11.58308576866979929352338495170, −11.10243148800942665394245225630, −9.668861736681283416309826444731, −8.744975007304773776853400418058, −7.56259890146580419679183122821, −6.30066415976652779133915936087, −5.10073904849151121515528887970, −4.43473851816862788064673651876, −3.42566783995476983091508702397, −1.56568752475378383043535673980, 2.54158487228008431246839753345, 3.55170658982895102797845505484, 4.84259991501080136095445023405, 5.95451606252530015226940757024, 6.46423356562691877185709349814, 7.75928506069318038345276459003, 8.899000064643455336545478570972, 10.13579057245338942299951987747, 11.17189891109373505345773899226, 11.97808821702612789459458414250

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