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Properties

Label 2-315-63.16-c1-0-9
Degree $2$
Conductor $315$
Sign $0.766 + 0.642i$
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  + (−1.04 − 1.80i)2-s + (0.853 + 1.50i)3-s + (−1.17 + 2.03i)4-s + 5-s + (1.83 − 3.11i)6-s + (2.35 − 1.21i)7-s + 0.738·8-s + (−1.54 + 2.57i)9-s + (−1.04 − 1.80i)10-s − 0.0154·11-s + (−4.07 − 0.0341i)12-s + (3.01 + 5.21i)13-s + (−4.64 − 2.97i)14-s + (0.853 + 1.50i)15-s + (1.58 + 2.74i)16-s + (−0.453 − 0.786i)17-s + ⋯
L(s)  = 1  + (−0.737 − 1.27i)2-s + (0.492 + 0.870i)3-s + (−0.588 + 1.01i)4-s + 0.447·5-s + (0.748 − 1.27i)6-s + (0.888 − 0.459i)7-s + 0.261·8-s + (−0.514 + 0.857i)9-s + (−0.329 − 0.571i)10-s − 0.00465·11-s + (−1.17 − 0.00986i)12-s + (0.834 + 1.44i)13-s + (−1.24 − 0.796i)14-s + (0.220 + 0.389i)15-s + (0.395 + 0.685i)16-s + (−0.110 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13146 - 0.411715i\)
\(L(\frac12)\) \(\approx\) \(1.13146 - 0.411715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.853 - 1.50i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.35 + 1.21i)T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 0.0154T + 11T^{2} \)
13 \( 1 + (-3.01 - 5.21i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.453 + 0.786i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.94 + 6.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.32T + 23T^{2} \)
29 \( 1 + (-1.08 + 1.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.610 - 1.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.79 - 8.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.46 + 9.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.234 + 0.406i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.53 - 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.39 + 2.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.60 - 4.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.12 + 7.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (0.466 + 0.808i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.51 + 6.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.41 - 14.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.33 - 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.11 + 5.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17753347766381060188007492314, −10.75132076335319501622380959890, −9.681796592618023568820472107665, −9.044718449658273284802266330167, −8.385962296801912192673080757877, −6.91814793538063503468792110367, −5.14273074183792479867130887573, −4.07588135919633506944997649321, −2.82632664623027923589757376223, −1.54659021923573602708311916147, 1.36494194220123796301599777354, 3.16517453443865349941817077323, 5.46183382367632974553116319119, 5.97310772982513360504656457659, 7.18861346585501436585349266331, 8.042760665191063211053015009959, 8.483359996821861149028420041067, 9.438993588531727295715650855281, 10.60850949886353398646904315999, 11.90891580638884039465957453378

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