↑

Properties

Label 2-315-63.4-c1-0-17
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

Related objects

Downloads

Learn more

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: $\chi_{315} (256, \cdot )$
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 \( 1 + (-3.11 - 5.39i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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