Properties

Label 2-315-9.4-c1-0-7
Degree $2$
Conductor $315$
Sign $0.782 - 0.622i$
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  + (0.746 + 1.29i)2-s + (−1.73 − 0.0231i)3-s + (−0.114 + 0.197i)4-s + (0.5 − 0.866i)5-s + (−1.26 − 2.25i)6-s + (−0.5 − 0.866i)7-s + 2.64·8-s + (2.99 + 0.0800i)9-s + 1.49·10-s + (2.28 + 3.95i)11-s + (0.202 − 0.339i)12-s + (1.66 − 2.89i)13-s + (0.746 − 1.29i)14-s + (−0.885 + 1.48i)15-s + (2.20 + 3.81i)16-s − 0.168·17-s + ⋯
L(s)  = 1  + (0.527 + 0.914i)2-s + (−0.999 − 0.0133i)3-s + (−0.0570 + 0.0987i)4-s + (0.223 − 0.387i)5-s + (−0.515 − 0.921i)6-s + (−0.188 − 0.327i)7-s + 0.935·8-s + (0.999 + 0.0266i)9-s + 0.472·10-s + (0.688 + 1.19i)11-s + (0.0583 − 0.0979i)12-s + (0.462 − 0.801i)13-s + (0.199 − 0.345i)14-s + (−0.228 + 0.384i)15-s + (0.550 + 0.953i)16-s − 0.0408·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41952 + 0.495316i\)
\(L(\frac12)\) \(\approx\) \(1.41952 + 0.495316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0231i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.746 - 1.29i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.28 - 3.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.66 + 2.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.168T + 17T^{2} \)
19 \( 1 - 3.03T + 19T^{2} \)
23 \( 1 + (-1.72 + 2.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.731 - 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.89 - 6.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.18T + 37T^{2} \)
41 \( 1 + (-4.73 + 8.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.79 - 3.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (2.49 - 4.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.76 + 6.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.25 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + (-1.32 - 2.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 4.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.53T + 89T^{2} \)
97 \( 1 + (1.59 + 2.76i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.00324678687383376204773422880, −10.64294901367829844611417794532, −10.20212277200872404414727292196, −8.885174758986394735321254702064, −7.31502392773909206222108685233, −6.89217120697862276019142514386, −5.74402897669941421125538178043, −5.04953649302334547847091295346, −4.01956401745477659940081619847, −1.40739149159242040001087936626, 1.50962758201706958086134527201, 3.19567666835508536729759667203, 4.21275234868909787525734408746, 5.54939681622524766976989446362, 6.44184540686408864484449972935, 7.50397657227417561538249915185, 9.035849191983495711836258235576, 10.06028993018635711903647174401, 11.10311293345221369546300194219, 11.49516941849396340688456124452

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