↑

Properties

Label 2-315-35.33-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.982 - 0.187i$
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  + (−0.650 − 2.42i)2-s + (−3.73 + 2.15i)4-s + (1.42 − 1.72i)5-s + (2.09 − 1.61i)7-s + (4.11 + 4.11i)8-s + (−5.11 − 2.33i)10-s + (−2.73 − 4.74i)11-s + (0.579 − 0.579i)13-s + (−5.29 − 4.02i)14-s + (3.00 − 5.19i)16-s + (−1.22 + 4.58i)17-s + (−0.220 + 0.381i)19-s + (−1.60 + 9.51i)20-s + (−9.73 + 9.73i)22-s + (−1.70 + 0.457i)23-s + ⋯
L(s)  = 1  + (−0.459 − 1.71i)2-s + (−1.86 + 1.07i)4-s + (0.636 − 0.770i)5-s + (0.791 − 0.611i)7-s + (1.45 + 1.45i)8-s + (−1.61 − 0.738i)10-s + (−0.825 − 1.42i)11-s + (0.160 − 0.160i)13-s + (−1.41 − 1.07i)14-s + (0.750 − 1.29i)16-s + (−0.298 + 1.11i)17-s + (−0.0505 + 0.0875i)19-s + (−0.358 + 2.12i)20-s + (−2.07 + 2.07i)22-s + (−0.355 + 0.0953i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.982 - 0.187i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.982 - 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0930743 + 0.981336i\)
\(L(\frac12)\) \(\approx\) \(0.0930743 + 0.981336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 1.72i)T \)
7 \( 1 + (-2.09 + 1.61i)T \)
good2 \( 1 + (0.650 + 2.42i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (2.73 + 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.579 + 0.579i)T - 13iT^{2} \)
17 \( 1 + (1.22 - 4.58i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.220 - 0.381i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.70 - 0.457i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.853iT - 29T^{2} \)
31 \( 1 + (-2.32 + 1.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0668 - 0.249i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.321iT - 41T^{2} \)
43 \( 1 + (0.631 + 0.631i)T + 43iT^{2} \)
47 \( 1 + (-7.91 + 2.12i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.96 - 11.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.89 + 5.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.73 + 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.16 - 1.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.79T + 71T^{2} \)
73 \( 1 + (-8.53 - 2.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.02 - 5.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.47 + 8.47i)T - 83iT^{2} \)
89 \( 1 + (-4.03 + 6.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.99 + 5.99i)T + 97iT^{2} \)
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

−10.79945857963541112123559705903, −10.65859632107634164453965885248, −9.498876344811458479514848467263, −8.488802512360520075067945475787, −8.051809371140356563269948053488, −5.91552283424091453873771208390, −4.71370907439386981708713318303, −3.59446990673227281482501085489, −2.13197340487467532639145944890, −0.873711586125560714847850052142, 2.29923239469866940796447999504, 4.69862567833152924927084657128, 5.41544821422332106868794156373, 6.51739129442689522524380720582, 7.31851836877031963695063542889, 8.081768834868037889014784924671, 9.234665624630198735702033655215, 9.866519992983761380126594016138, 10.94639363783983556616130181076, 12.25360944791139004621716331086

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