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Properties

Label 2-315-63.4-c1-0-5
Degree $2$
Conductor $315$
Sign $-0.350 - 0.936i$
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  + (−1.08 + 1.87i)2-s + (−1.55 − 0.752i)3-s + (−1.34 − 2.32i)4-s − 5-s + (3.09 − 2.10i)6-s + (2.14 − 1.54i)7-s + 1.47·8-s + (1.86 + 2.34i)9-s + (1.08 − 1.87i)10-s + 2.52·11-s + (0.343 + 4.63i)12-s + (−0.542 + 0.939i)13-s + (0.565 + 5.69i)14-s + (1.55 + 0.752i)15-s + (1.08 − 1.88i)16-s + (−2.85 + 4.93i)17-s + ⋯
L(s)  = 1  + (−0.764 + 1.32i)2-s + (−0.900 − 0.434i)3-s + (−0.670 − 1.16i)4-s − 0.447·5-s + (1.26 − 0.860i)6-s + (0.812 − 0.583i)7-s + 0.520·8-s + (0.622 + 0.782i)9-s + (0.342 − 0.592i)10-s + 0.760·11-s + (0.0992 + 1.33i)12-s + (−0.150 + 0.260i)13-s + (0.151 + 1.52i)14-s + (0.402 + 0.194i)15-s + (0.271 − 0.470i)16-s + (−0.691 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342373 + 0.493828i\)
\(L(\frac12)\) \(\approx\) \(0.342373 + 0.493828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.55 + 0.752i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.14 + 1.54i)T \)
good2 \( 1 + (1.08 - 1.87i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 + (0.542 - 0.939i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.85 - 4.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.75 - 3.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.273T + 23T^{2} \)
29 \( 1 + (-4.63 - 8.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.884 - 1.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.14 - 1.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.65 + 4.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.14 - 5.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.34 - 4.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.86 + 8.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.47 - 4.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.42 - 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + (0.107 - 0.186i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.95 + 3.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.18 - 8.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.78 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.91 + 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.83724828361085150050887077102, −10.94729014432889310251288420638, −10.04598914005902507976884099788, −8.703342489590456172265485795574, −7.978998229733859229843400439335, −7.07427349324897253126440137800, −6.44738759642578136559958391716, −5.31824974222375586564183499287, −4.18502308050638941528948482710, −1.27029919094346066735918247526, 0.75534106796348791503347894408, 2.50430082330798119689813764479, 4.03364974704191354500826850724, 5.06391313887567215856866150533, 6.45721200794204606119470273872, 7.84914732700760813147697770827, 9.038262431961835343170444714457, 9.580882577663408498658627520349, 10.69044321532931493246660433605, 11.47408370335841364231731071522

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