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Properties

Label 2-315-63.16-c1-0-18
Degree $2$
Conductor $315$
Sign $0.949 - 0.312i$
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  + (0.195 + 0.338i)2-s + (0.811 + 1.53i)3-s + (0.923 − 1.59i)4-s + 5-s + (−0.359 + 0.573i)6-s + (0.0590 − 2.64i)7-s + 1.50·8-s + (−1.68 + 2.48i)9-s + (0.195 + 0.338i)10-s + 3.11·11-s + (3.19 + 0.114i)12-s + (−1.07 − 1.86i)13-s + (0.907 − 0.497i)14-s + (0.811 + 1.53i)15-s + (−1.55 − 2.68i)16-s + (−0.0261 − 0.0453i)17-s + ⋯
L(s)  = 1  + (0.138 + 0.239i)2-s + (0.468 + 0.883i)3-s + (0.461 − 0.799i)4-s + 0.447·5-s + (−0.146 + 0.234i)6-s + (0.0223 − 0.999i)7-s + 0.531·8-s + (−0.560 + 0.827i)9-s + (0.0618 + 0.107i)10-s + 0.939·11-s + (0.922 + 0.0331i)12-s + (−0.298 − 0.517i)13-s + (0.242 − 0.132i)14-s + (0.209 + 0.395i)15-s + (−0.388 − 0.672i)16-s + (−0.00634 − 0.0109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.949 - 0.312i$
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: \((2,\ 315,\ (\ :1/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88089 + 0.301860i\)
\(L(\frac12)\) \(\approx\) \(1.88089 + 0.301860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.811 - 1.53i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.0590 + 2.64i)T \)
good2 \( 1 + (-0.195 - 0.338i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 - 3.11T + 11T^{2} \)
13 \( 1 + (1.07 + 1.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0261 + 0.0453i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.73 - 6.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.105T + 23T^{2} \)
29 \( 1 + (2.27 - 3.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.298 - 0.517i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.88 - 8.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.29 + 5.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.63 + 6.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.39 + 4.13i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.625 - 1.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.42 - 2.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.22T + 71T^{2} \)
73 \( 1 + (2.38 + 4.13i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.00 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.38 - 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.22 + 7.31i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.575 + 0.996i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36282412671116010747209636289, −10.47600328491484921704549248192, −10.07852890547305933221979410982, −9.095362930609732676175295177599, −7.87225374023230631023454664720, −6.75598302342412128295791769905, −5.71370455550834517454510156576, −4.63077957943380098978172661194, −3.49904991899549072235147234498, −1.72682735705682361736853514807, 1.96954358771926474801728998925, 2.75734622354358788409335933744, 4.23146422530664181745000909130, 5.99713434334347904765040393147, 6.78880121337451527284744500282, 7.74471859317057533566356455135, 8.883005619798645855598024121733, 9.328108592894887066252946244966, 11.14610810308102299931822072603, 11.71708155379385578027930539265

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