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Properties

Label 2-315-63.20-c1-0-20
Degree $2$
Conductor $315$
Sign $0.919 - 0.393i$
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  + (1.94 + 1.12i)2-s + (0.913 − 1.47i)3-s + (1.52 + 2.64i)4-s + (−0.5 − 0.866i)5-s + (3.43 − 1.83i)6-s + (0.315 + 2.62i)7-s + 2.38i·8-s + (−1.33 − 2.68i)9-s − 2.24i·10-s + (2.13 + 1.23i)11-s + (5.29 + 0.170i)12-s + (−0.336 + 0.194i)13-s + (−2.33 + 5.47i)14-s + (−1.73 − 0.0555i)15-s + (0.378 − 0.655i)16-s − 4.81·17-s + ⋯
L(s)  = 1  + (1.37 + 0.795i)2-s + (0.527 − 0.849i)3-s + (0.764 + 1.32i)4-s + (−0.223 − 0.387i)5-s + (1.40 − 0.750i)6-s + (0.119 + 0.992i)7-s + 0.842i·8-s + (−0.443 − 0.896i)9-s − 0.711i·10-s + (0.643 + 0.371i)11-s + (1.52 + 0.0490i)12-s + (−0.0933 + 0.0538i)13-s + (−0.625 + 1.46i)14-s + (−0.446 − 0.0143i)15-s + (0.0946 − 0.163i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.919 - 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85080 + 0.584457i\)
\(L(\frac12)\) \(\approx\) \(2.85080 + 0.584457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 + 1.47i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.315 - 2.62i)T \)
good2 \( 1 + (-1.94 - 1.12i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.13 - 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.336 - 0.194i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.81T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 + (1.85 - 1.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.02 + 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.61 - 3.24i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 + (-0.476 - 0.825i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.21 - 9.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.81 + 4.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.72iT - 53T^{2} \)
59 \( 1 + (-6.18 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.631 + 0.364i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.73 - 9.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.1iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + (-7.35 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.97 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (-3.46 - 2.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.99508624358986171621864085991, −11.53617026233126686368726647057, −9.451865929506711523132317898122, −8.637262413531770344005980509928, −7.59616502591241408262106875829, −6.68736804260844743694649678636, −5.87939680201641472574067276774, −4.74350188177208990959711160820, −3.57101082704019994088135833782, −2.15376984188144405615389485512, 2.20352747723516375663676607121, 3.61546241275800817310047984749, 4.04242110925046608889025679112, 5.12166109182287196611444333900, 6.42082573924649084970557461679, 7.74156712329120475909438405993, 9.003407562952187331756873513035, 10.14550063632330495065316612700, 11.00288532682512937429750603450, 11.33740639872328973954113463010

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