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Properties

Label 2-315-315.23-c1-0-38
Degree $2$
Conductor $315$
Sign $-0.938 - 0.346i$
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  + (−0.477 − 0.477i)2-s + (−1.72 + 0.114i)3-s − 1.54i·4-s + (0.814 − 2.08i)5-s + (0.879 + 0.770i)6-s + (−2.51 − 0.835i)7-s + (−1.69 + 1.69i)8-s + (2.97 − 0.396i)9-s + (−1.38 + 0.604i)10-s + (0.137 + 0.0791i)11-s + (0.177 + 2.66i)12-s + (−0.549 + 0.147i)13-s + (0.799 + 1.59i)14-s + (−1.16 + 3.69i)15-s − 1.47·16-s + (−1.11 + 4.15i)17-s + ⋯
L(s)  = 1  + (−0.337 − 0.337i)2-s + (−0.997 + 0.0662i)3-s − 0.772i·4-s + (0.364 − 0.931i)5-s + (0.359 + 0.314i)6-s + (−0.948 − 0.315i)7-s + (−0.598 + 0.598i)8-s + (0.991 − 0.132i)9-s + (−0.437 + 0.191i)10-s + (0.0413 + 0.0238i)11-s + (0.0511 + 0.770i)12-s + (−0.152 + 0.0408i)13-s + (0.213 + 0.426i)14-s + (−0.301 + 0.953i)15-s − 0.368·16-s + (−0.269 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.938 - 0.346i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.938 - 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0604625 + 0.338414i\)
\(L(\frac12)\) \(\approx\) \(0.0604625 + 0.338414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.114i)T \)
5 \( 1 + (-0.814 + 2.08i)T \)
7 \( 1 + (2.51 + 0.835i)T \)
good2 \( 1 + (0.477 + 0.477i)T + 2iT^{2} \)
11 \( 1 + (-0.137 - 0.0791i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.549 - 0.147i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.11 - 4.15i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.32 + 1.91i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.02 + 0.274i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.59 - 4.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.05T + 31T^{2} \)
37 \( 1 + (3.10 + 11.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.20 + 2.42i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.495 - 1.84i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.76 + 2.76i)T + 47iT^{2} \)
53 \( 1 + (-12.4 - 3.32i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 + (-7.28 + 7.28i)T - 67iT^{2} \)
71 \( 1 + 9.09iT - 71T^{2} \)
73 \( 1 + (-1.02 + 3.81i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + 6.24iT - 79T^{2} \)
83 \( 1 + (16.4 + 4.40i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.89 + 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.7 + 3.67i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.83785094376078803783634527629, −10.36313839925119412621163078317, −9.449780775874043700674096714613, −8.705087319857529190162611540337, −6.95889035222435617735410378465, −6.05927329316904545989072293479, −5.27827640077679764641252355051, −4.08963358746389035318355626869, −1.83097716906424958707324915337, −0.29894157482916625129089241017, 2.63274320416627942168656410548, 3.93079960175094246515468247760, 5.56222885696627081837823792238, 6.68651523859810362622138669390, 6.94191094485315776560296218742, 8.289029645217386971749812263403, 9.610998494832500066179124070503, 10.15649715509416057662046290475, 11.39030429126467330502596700993, 12.03304824570788727685839260164

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