↑

Properties

Label 2-315-35.17-c1-0-15
Degree $2$
Conductor $315$
Sign $-0.670 + 0.741i$
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  + (0.401 − 1.49i)2-s + (−0.346 − 0.200i)4-s + (−2.14 + 0.625i)5-s + (−1.01 − 2.44i)7-s + (1.75 − 1.75i)8-s + (0.0747 + 3.46i)10-s + (2.59 − 4.49i)11-s + (−3.30 − 3.30i)13-s + (−4.06 + 0.545i)14-s + (−2.32 − 4.01i)16-s + (−0.00519 − 0.0194i)17-s + (1.24 + 2.14i)19-s + (0.869 + 0.213i)20-s + (−5.68 − 5.68i)22-s + (2.24 + 0.601i)23-s + ⋯
L(s)  = 1  + (0.283 − 1.05i)2-s + (−0.173 − 0.100i)4-s + (−0.960 + 0.279i)5-s + (−0.385 − 0.922i)7-s + (0.619 − 0.619i)8-s + (0.0236 + 1.09i)10-s + (0.782 − 1.35i)11-s + (−0.917 − 0.917i)13-s + (−1.08 + 0.145i)14-s + (−0.580 − 1.00i)16-s + (−0.00126 − 0.00470i)17-s + (0.284 + 0.492i)19-s + (0.194 + 0.0476i)20-s + (−1.21 − 1.21i)22-s + (0.468 + 0.125i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.670 + 0.741i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531576 - 1.19771i\)
\(L(\frac12)\) \(\approx\) \(0.531576 - 1.19771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 0.625i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
good2 \( 1 + (-0.401 + 1.49i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.59 + 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.30 + 3.30i)T + 13iT^{2} \)
17 \( 1 + (0.00519 + 0.0194i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.24 - 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.24 - 0.601i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 - 3.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.714 - 2.66i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (2.79 - 2.79i)T - 43iT^{2} \)
47 \( 1 + (1.13 + 0.303i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.23 + 4.60i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.222 + 0.385i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 0.684i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.70 + 1.52i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.14T + 71T^{2} \)
73 \( 1 + (-7.15 + 1.91i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.47 - 2.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.77 - 3.77i)T + 83iT^{2} \)
89 \( 1 + (1.91 + 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.5 + 10.5i)T - 97iT^{2} \)
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

−11.27747053038583472873458401734, −10.67551567806904609511498232861, −9.879370740605899581845795940062, −8.470619162813195604246644648201, −7.41928395719362379496288767361, −6.62534579957153646485234788697, −4.86662225485722976024095969230, −3.54085249481670711288865475323, −3.16252209596247066674332864596, −0.914446198487711909974139228954, 2.28699438515549865844280094204, 4.26626153799601902733152104039, 4.97543404274395724791972838005, 6.34900924775785648690499056041, 7.08521035921552712573718297381, 7.922930139374948401331267928759, 9.044807232498586580607372003075, 9.862307035753412643129493001703, 11.50925714206776115581730683018, 11.90326950205404891928271845652

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