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Properties

Label 2-315-63.41-c1-0-1
Degree $2$
Conductor $315$
Sign $-0.858 - 0.512i$
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  + (−2.29 + 1.32i)2-s + (−1.70 + 0.281i)3-s + (2.50 − 4.33i)4-s + (−0.5 + 0.866i)5-s + (3.54 − 2.90i)6-s + (−1.33 − 2.28i)7-s + 7.94i·8-s + (2.84 − 0.963i)9-s − 2.64i·10-s + (0.416 − 0.240i)11-s + (−3.05 + 8.11i)12-s + (2.11 + 1.22i)13-s + (6.08 + 3.45i)14-s + (0.610 − 1.62i)15-s + (−5.51 − 9.54i)16-s + 1.67·17-s + ⋯
L(s)  = 1  + (−1.62 + 0.935i)2-s + (−0.986 + 0.162i)3-s + (1.25 − 2.16i)4-s + (−0.223 + 0.387i)5-s + (1.44 − 1.18i)6-s + (−0.506 − 0.862i)7-s + 2.81i·8-s + (0.947 − 0.321i)9-s − 0.836i·10-s + (0.125 − 0.0724i)11-s + (−0.881 + 2.34i)12-s + (0.587 + 0.339i)13-s + (1.62 + 0.924i)14-s + (0.157 − 0.418i)15-s + (−1.37 − 2.38i)16-s + 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.858 - 0.512i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.858 - 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0698905 + 0.253193i\)
\(L(\frac12)\) \(\approx\) \(0.0698905 + 0.253193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.281i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.33 + 2.28i)T \)
good2 \( 1 + (2.29 - 1.32i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-0.416 + 0.240i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.11 - 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.67T + 17T^{2} \)
19 \( 1 - 7.56iT - 19T^{2} \)
23 \( 1 + (2.87 + 1.66i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.15 - 1.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.09 + 4.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.34T + 37T^{2} \)
41 \( 1 + (4.37 - 7.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.73 - 9.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.433 - 0.751i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.15iT - 53T^{2} \)
59 \( 1 + (4.89 - 8.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.60 + 4.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.821 - 1.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.44iT - 71T^{2} \)
73 \( 1 - 9.10iT - 73T^{2} \)
79 \( 1 + (-5.20 - 9.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.09 + 5.35i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (12.3 - 7.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.47237658517811763400116595304, −10.80393909012888825106980615751, −10.04252334951904528617284131834, −9.446926422939240528209274376079, −8.018308917498462219424989705248, −7.33710185595462955524651752473, −6.34568527361087301833676428200, −5.80827516768304708181515012363, −3.99613559706506275533249798236, −1.29227883747860035814033111751, 0.40250730986382327150483269528, 2.01392976432202028617790322508, 3.58601146360788354437976996415, 5.39255379134666705418992236984, 6.71332646121684882530445829745, 7.63980378566750003810714802381, 8.808307465765863140198873476495, 9.387742114971185991382320814690, 10.43309915746862869335345584805, 11.18230345472990090538173867278

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