Properties

Label 2-315-21.20-c1-0-7
Degree $2$
Conductor $315$
Sign $-0.974 + 0.225i$
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  − 1.93i·2-s − 1.73·4-s − 5-s + (1 − 2.44i)7-s − 0.517i·8-s + 1.93i·10-s − 5.27i·11-s + 6.69i·13-s + (−4.73 − 1.93i)14-s − 4.46·16-s − 3.46·17-s − 6.69i·19-s + 1.73·20-s − 10.1·22-s + 1.41i·23-s + ⋯
L(s)  = 1  − 1.36i·2-s − 0.866·4-s − 0.447·5-s + (0.377 − 0.925i)7-s − 0.183i·8-s + 0.610i·10-s − 1.59i·11-s + 1.85i·13-s + (−1.26 − 0.516i)14-s − 1.11·16-s − 0.840·17-s − 1.53i·19-s + 0.387·20-s − 2.17·22-s + 0.294i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129829 - 1.13450i\)
\(L(\frac12)\) \(\approx\) \(0.129829 - 1.13450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (-1 + 2.44i)T \)
good2 \( 1 + 1.93iT - 2T^{2} \)
11 \( 1 + 5.27iT - 11T^{2} \)
13 \( 1 - 6.69iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 6.69iT - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 1.79iT - 31T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 - 4.52iT - 53T^{2} \)
59 \( 1 - 2.53T + 59T^{2} \)
61 \( 1 + 3.58iT - 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 - 9.41iT - 71T^{2} \)
73 \( 1 + 6.69iT - 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−11.12377364491127807752431291750, −10.89990576946031219567400863911, −9.399118480375941869621998720136, −8.814570820759802067482863128329, −7.38450281613412684332292429467, −6.42599872617357549692423868531, −4.53115895104720968665450312972, −3.87004547748875191951105188808, −2.50460288368402098040666663796, −0.854850235103496646194412884209, 2.41677867244670426775587416637, 4.35825056307535366849534896428, 5.39064607497294255953299002215, 6.19491046902529440168830119634, 7.51398810241649554935762631301, 7.941245228743110013209012295908, 8.912688762507248293230162214516, 10.06713061858503804127670437109, 11.18014970939378780546480439397, 12.36265677901363705702187679190

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