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Properties

Label 2-315-63.5-c1-0-21
Degree $2$
Conductor $315$
Sign $0.915 - 0.402i$
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  + 1.23i·2-s + (1.51 − 0.843i)3-s + 0.470·4-s + (−0.5 − 0.866i)5-s + (1.04 + 1.87i)6-s + (2.44 − 1.02i)7-s + 3.05i·8-s + (1.57 − 2.55i)9-s + (1.07 − 0.618i)10-s + (−1.53 − 0.884i)11-s + (0.712 − 0.397i)12-s + (−5.15 − 2.97i)13-s + (1.26 + 3.01i)14-s + (−1.48 − 0.888i)15-s − 2.83·16-s + (2.04 + 3.53i)17-s + ⋯
L(s)  = 1  + 0.874i·2-s + (0.873 − 0.487i)3-s + 0.235·4-s + (−0.223 − 0.387i)5-s + (0.425 + 0.763i)6-s + (0.922 − 0.386i)7-s + 1.08i·8-s + (0.525 − 0.850i)9-s + (0.338 − 0.195i)10-s + (−0.462 − 0.266i)11-s + (0.205 − 0.114i)12-s + (−1.43 − 0.826i)13-s + (0.337 + 0.806i)14-s + (−0.383 − 0.229i)15-s − 0.709·16-s + (0.495 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.915 - 0.402i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88978 + 0.397480i\)
\(L(\frac12)\) \(\approx\) \(1.88978 + 0.397480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 0.843i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.44 + 1.02i)T \)
good2 \( 1 - 1.23iT - 2T^{2} \)
11 \( 1 + (1.53 + 0.884i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.15 + 2.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.04 - 3.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.71 - 2.72i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.08 - 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.244iT - 31T^{2} \)
37 \( 1 + (2.29 - 3.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.718 + 1.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.63 - 6.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.0152T + 47T^{2} \)
53 \( 1 + (6.05 - 3.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.828T + 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (4.15 - 2.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.06T + 79T^{2} \)
83 \( 1 + (-2.03 - 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.64 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.65 - 2.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.93149858893563948618052502153, −10.76915550358977383558217287675, −9.658124567521053212379837979007, −8.379934247452930988621457104543, −7.67814596549180883907869314128, −7.43224500044633312828132254587, −5.86864104153361408997222833741, −4.89191906231633548365745850436, −3.24469793744705535919684152050, −1.73719578070069047012978063059, 2.06932592624033240710832525234, 2.81665711180819325405254967712, 4.17210558075081460097099527918, 5.24333775534361914325962112412, 7.26360739964731939393657342663, 7.61211830220274131914234394778, 9.155396851865540261361729147237, 9.776030297115356501934478346025, 10.68894088590024324020707476824, 11.62285105199554053837363087127

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