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Properties

Label 2-315-63.16-c1-0-23
Degree $2$
Conductor $315$
Sign $0.576 + 0.817i$
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  + (0.154 + 0.266i)2-s + (1.05 − 1.37i)3-s + (0.952 − 1.64i)4-s − 5-s + (0.529 + 0.0689i)6-s + (2.20 + 1.46i)7-s + 1.20·8-s + (−0.783 − 2.89i)9-s + (−0.154 − 0.266i)10-s − 1.10·11-s + (−1.26 − 3.04i)12-s + (2.54 + 4.40i)13-s + (−0.0508 + 0.813i)14-s + (−1.05 + 1.37i)15-s + (−1.71 − 2.97i)16-s + (−3.17 − 5.50i)17-s + ⋯
L(s)  = 1  + (0.108 + 0.188i)2-s + (0.607 − 0.794i)3-s + (0.476 − 0.824i)4-s − 0.447·5-s + (0.216 + 0.0281i)6-s + (0.833 + 0.553i)7-s + 0.425·8-s + (−0.261 − 0.965i)9-s + (−0.0487 − 0.0843i)10-s − 0.331·11-s + (−0.365 − 0.879i)12-s + (0.705 + 1.22i)13-s + (−0.0135 + 0.217i)14-s + (−0.271 + 0.355i)15-s + (−0.429 − 0.744i)16-s + (−0.770 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.576 + 0.817i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.576 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61106 - 0.835054i\)
\(L(\frac12)\) \(\approx\) \(1.61106 - 0.835054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 1.46i)T \)
good2 \( 1 + (-0.154 - 0.266i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 1.10T + 11T^{2} \)
13 \( 1 + (-2.54 - 4.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.17 + 5.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.518 + 0.897i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.372T + 23T^{2} \)
29 \( 1 + (1.91 - 3.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.11 - 8.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.42 + 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.62 - 8.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.67 + 4.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.68 - 8.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.35 - 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.33 + 4.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.15 - 7.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.16 - 2.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 + (5.94 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.26 + 9.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.01 - 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.40 - 5.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.00 + 3.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.44988496023841947615974713607, −10.95133791899662721123211998433, −9.307326922036224329493831341476, −8.739158160891540019449811650879, −7.46316590062154868766582210295, −6.85599095656482535616681094459, −5.68614338684269919440326351238, −4.48643239617733358600730433240, −2.66994038418280432831061928690, −1.47394615361611512375634585328, 2.24149005761085129420026662001, 3.67592707355012629836604548883, 4.21576846901257054644337752157, 5.70938224878508920810860152822, 7.42837969378350139867669312495, 8.068681072587668014088623742503, 8.676755616378001303931572669049, 10.24014351551260127373581733800, 10.91142606407393764499468543339, 11.49306008493655211991634841410

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