↑

Properties

Label 2-315-105.2-c1-0-13
Degree $2$
Conductor $315$
Sign $-0.886 + 0.463i$
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  + (−0.494 − 1.84i)2-s + (−1.43 + 0.826i)4-s + (0.541 − 2.16i)5-s + (2.64 − 0.0471i)7-s + (−0.468 − 0.468i)8-s + (−4.27 + 0.0725i)10-s + (0.971 − 0.561i)11-s + (0.830 − 0.830i)13-s + (−1.39 − 4.86i)14-s + (−2.28 + 3.96i)16-s + (2.16 + 0.578i)17-s + (−5.19 − 3.00i)19-s + (1.01 + 3.55i)20-s + (−1.51 − 1.51i)22-s + (0.641 − 0.171i)23-s + ⋯
L(s)  = 1  + (−0.349 − 1.30i)2-s + (−0.715 + 0.413i)4-s + (0.242 − 0.970i)5-s + (0.999 − 0.0178i)7-s + (−0.165 − 0.165i)8-s + (−1.35 + 0.0229i)10-s + (0.293 − 0.169i)11-s + (0.230 − 0.230i)13-s + (−0.372 − 1.29i)14-s + (−0.571 + 0.990i)16-s + (0.523 + 0.140i)17-s + (−1.19 − 0.688i)19-s + (0.227 + 0.794i)20-s + (−0.323 − 0.323i)22-s + (0.133 − 0.0358i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.886 + 0.463i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.886 + 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285191 - 1.16080i\)
\(L(\frac12)\) \(\approx\) \(0.285191 - 1.16080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.541 + 2.16i)T \)
7 \( 1 + (-2.64 + 0.0471i)T \)
good2 \( 1 + (0.494 + 1.84i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.971 + 0.561i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.830 + 0.830i)T - 13iT^{2} \)
17 \( 1 + (-2.16 - 0.578i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.19 + 3.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.641 + 0.171i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.88T + 29T^{2} \)
31 \( 1 + (-4.19 - 7.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.51 + 1.47i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.31iT - 41T^{2} \)
43 \( 1 + (-5.26 + 5.26i)T - 43iT^{2} \)
47 \( 1 + (-2.55 - 9.52i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.211 + 0.790i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.75 - 3.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.57 + 4.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 6.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + (4.49 + 1.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.12 - 5.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.5 - 11.5i)T + 83iT^{2} \)
89 \( 1 + (-1.76 + 3.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.6 - 10.6i)T + 97iT^{2} \)
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

−11.15846198217220405695349501669, −10.59455250385063395183548124932, −9.390007724802954480180092245368, −8.798225079525620101200731414217, −7.83524308110510254191795979912, −6.19235908373438282496738322380, −4.92643389300017796479781231262, −3.85193269002316705458202892584, −2.21594261549918780028833326637, −1.05890149479298471405614960692, 2.22950863884669519860102337695, 4.06841232542458750119293363683, 5.55366283543383302793625847404, 6.29192161548858702862771008973, 7.34335846279774840745680444996, 7.958752769378978805605994986451, 8.984432142111374788677125319955, 10.02855516337740173652769231611, 11.14145976499892679973716520663, 11.78904760187755004333681031151

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