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Properties

Label 2-315-35.33-c1-0-15
Degree $2$
Conductor $315$
Sign $-0.410 + 0.911i$
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  + (−0.259 − 0.969i)2-s + (0.859 − 0.496i)4-s + (1.40 − 1.73i)5-s + (−1.06 − 2.42i)7-s + (−2.12 − 2.12i)8-s + (−2.05 − 0.910i)10-s + (1.78 + 3.08i)11-s + (−2.78 + 2.78i)13-s + (−2.07 + 1.65i)14-s + (−0.514 + 0.891i)16-s + (0.135 − 0.506i)17-s + (2.06 − 3.57i)19-s + (0.344 − 2.19i)20-s + (2.53 − 2.53i)22-s + (2.49 − 0.668i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.685i)2-s + (0.429 − 0.248i)4-s + (0.628 − 0.777i)5-s + (−0.401 − 0.915i)7-s + (−0.750 − 0.750i)8-s + (−0.648 − 0.287i)10-s + (0.537 + 0.931i)11-s + (−0.772 + 0.772i)13-s + (−0.554 + 0.443i)14-s + (−0.128 + 0.222i)16-s + (0.0329 − 0.122i)17-s + (0.473 − 0.819i)19-s + (0.0770 − 0.490i)20-s + (0.539 − 0.539i)22-s + (0.520 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.410 + 0.911i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749461 - 1.15951i\)
\(L(\frac12)\) \(\approx\) \(0.749461 - 1.15951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.73i)T \)
7 \( 1 + (1.06 + 2.42i)T \)
good2 \( 1 + (0.259 + 0.969i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.78 - 3.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.78 - 2.78i)T - 13iT^{2} \)
17 \( 1 + (-0.135 + 0.506i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.06 + 3.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.49 + 0.668i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.14iT - 29T^{2} \)
31 \( 1 + (1.71 - 0.988i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0409 + 0.152i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.28iT - 41T^{2} \)
43 \( 1 + (-9.01 - 9.01i)T + 43iT^{2} \)
47 \( 1 + (-5.19 + 1.39i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.48 + 5.55i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.67 - 5.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.32 - 1.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 + (14.8 + 3.98i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.1 - 7.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.42 - 9.42i)T - 83iT^{2} \)
89 \( 1 + (5.52 - 9.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 + 2.48i)T + 97iT^{2} \)
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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

−11.31884161261258757572108328189, −10.35750297081134147410326188829, −9.560327255435376971802821926622, −9.077899122711887933435510631585, −7.23193718084353866043126089458, −6.69623353277591275508844261626, −5.23549611242332896772822357556, −4.07229768944412820148125142235, −2.44723164817621967731561452738, −1.13328877028388839118231622629, 2.41293824172093781888951611008, 3.33565051332990570748122156609, 5.62335334754353093542169757287, 6.00903460886765361986385643416, 7.07868329159642660791064071167, 8.028500318227425424930053507393, 9.063242189385255983140288631593, 9.963201538753948785683049619919, 11.09082860946079057530610155471, 11.88861218548237197138185354713

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