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Properties

Label 2-315-105.2-c1-0-11
Degree $2$
Conductor $315$
Sign $0.975 + 0.221i$
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  + (0.271 + 1.01i)2-s + (0.777 − 0.449i)4-s + (−0.310 − 2.21i)5-s + (−1.26 − 2.32i)7-s + (2.15 + 2.15i)8-s + (2.16 − 0.915i)10-s + (−1.52 + 0.880i)11-s + (4.98 − 4.98i)13-s + (2.00 − 1.91i)14-s + (−0.698 + 1.20i)16-s + (0.458 + 0.122i)17-s + (2.33 + 1.34i)19-s + (−1.23 − 1.58i)20-s + (−1.30 − 1.30i)22-s + (−6.30 + 1.69i)23-s + ⋯
L(s)  = 1  + (0.192 + 0.716i)2-s + (0.388 − 0.224i)4-s + (−0.138 − 0.990i)5-s + (−0.479 − 0.877i)7-s + (0.760 + 0.760i)8-s + (0.683 − 0.289i)10-s + (−0.460 + 0.265i)11-s + (1.38 − 1.38i)13-s + (0.537 − 0.512i)14-s + (−0.174 + 0.302i)16-s + (0.111 + 0.0298i)17-s + (0.535 + 0.309i)19-s + (−0.276 − 0.354i)20-s + (−0.278 − 0.278i)22-s + (−1.31 + 0.352i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57969 - 0.176914i\)
\(L(\frac12)\) \(\approx\) \(1.57969 - 0.176914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.310 + 2.21i)T \)
7 \( 1 + (1.26 + 2.32i)T \)
good2 \( 1 + (-0.271 - 1.01i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.52 - 0.880i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.98 + 4.98i)T - 13iT^{2} \)
17 \( 1 + (-0.458 - 0.122i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.33 - 1.34i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.30 - 1.69i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.01T + 29T^{2} \)
31 \( 1 + (-3.95 - 6.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.45 - 1.73i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.20iT - 41T^{2} \)
43 \( 1 + (2.28 - 2.28i)T - 43iT^{2} \)
47 \( 1 + (0.393 + 1.47i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.26 + 8.45i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.17 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.530 - 1.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.79iT - 71T^{2} \)
73 \( 1 + (1.45 + 0.391i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.08 - 5.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.07 + 2.07i)T + 83iT^{2} \)
89 \( 1 + (-1.64 + 2.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 - 3.21i)T + 97iT^{2} \)
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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

−11.67932262113555821871673900200, −10.49544957676672583728200411369, −9.988232719943967282633743480105, −8.320049141380480400144845768333, −7.925583209088179338147844091245, −6.67211299310288696271067212692, −5.73857599854728911474070555866, −4.79592171170903722118959883312, −3.40097586181919338833783581487, −1.23970888360901935907712918238, 2.09978065020531732209136057919, 3.11624630890626801842783842473, 4.12498455215363641931023928575, 6.03342208055061291006645275815, 6.66632728482469619186563195690, 7.85107246697862720193928535692, 8.999239067779847924113248319607, 10.15795850305528390079727311120, 10.88051819467080072603307367210, 11.80596039447486639895475898022

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