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Properties

Label 2-315-35.33-c1-0-7
Degree $2$
Conductor $315$
Sign $0.997 + 0.0696i$
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  + (−0.105 − 0.394i)2-s + (1.58 − 0.916i)4-s + (0.699 + 2.12i)5-s + (2.57 + 0.605i)7-s + (−1.10 − 1.10i)8-s + (0.763 − 0.500i)10-s + (0.463 + 0.803i)11-s + (−4.08 + 4.08i)13-s + (−0.0332 − 1.08i)14-s + (1.51 − 2.62i)16-s + (0.192 − 0.719i)17-s + (1.21 − 2.11i)19-s + (3.05 + 2.73i)20-s + (0.267 − 0.267i)22-s + (5.00 − 1.34i)23-s + ⋯
L(s)  = 1  + (−0.0747 − 0.278i)2-s + (0.793 − 0.458i)4-s + (0.312 + 0.949i)5-s + (0.973 + 0.228i)7-s + (−0.391 − 0.391i)8-s + (0.241 − 0.158i)10-s + (0.139 + 0.242i)11-s + (−1.13 + 1.13i)13-s + (−0.00889 − 0.288i)14-s + (0.378 − 0.655i)16-s + (0.0467 − 0.174i)17-s + (0.279 − 0.484i)19-s + (0.683 + 0.610i)20-s + (0.0571 − 0.0571i)22-s + (1.04 − 0.279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0696i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.997 + 0.0696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64195 - 0.0572681i\)
\(L(\frac12)\) \(\approx\) \(1.64195 - 0.0572681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.699 - 2.12i)T \)
7 \( 1 + (-2.57 - 0.605i)T \)
good2 \( 1 + (0.105 + 0.394i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.463 - 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.08 - 4.08i)T - 13iT^{2} \)
17 \( 1 + (-0.192 + 0.719i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.21 + 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.00 + 1.34i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 - 0.607i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.472 + 1.76i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.97iT - 41T^{2} \)
43 \( 1 + (0.781 + 0.781i)T + 43iT^{2} \)
47 \( 1 + (10.0 - 2.70i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.72 - 6.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.91 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.72 + 2.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 + 2.80i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + (-4.02 - 1.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.02 + 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.91 - 5.91i)T - 83iT^{2} \)
89 \( 1 + (-7.78 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−11.55268901003037147725273469660, −10.84777064152703239772662314130, −9.928833036364046126250745063857, −9.102442703554212124185842846481, −7.48274143709486573372399418650, −6.93827974509376359455434374608, −5.81102087625793633516041278341, −4.58963437502501491568746257022, −2.78093273134003410639609835210, −1.86608884974410204068524600472, 1.56811354202568233431340486637, 3.15383810847233591277885776138, 4.84599027188599359308420018151, 5.60105401454901371849993683528, 7.00434128513840437410287003622, 7.936624350884000493333018732800, 8.537724724070731528441956020477, 9.794318287678427233291617475506, 10.84720089455237128098751797775, 11.72976479021819457077580779376

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