↑

Properties

Label 2-315-63.59-c1-0-29
Degree $2$
Conductor $315$
Sign $-0.879 - 0.476i$
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  + (−0.948 − 0.547i)2-s + (−0.237 − 1.71i)3-s + (−0.399 − 0.692i)4-s − 5-s + (−0.714 + 1.75i)6-s + (1.38 − 2.25i)7-s + 3.06i·8-s + (−2.88 + 0.816i)9-s + (0.948 + 0.547i)10-s − 5.93i·11-s + (−1.09 + 0.850i)12-s + (−0.902 − 0.521i)13-s + (−2.54 + 1.38i)14-s + (0.237 + 1.71i)15-s + (0.880 − 1.52i)16-s + (−3.21 + 5.56i)17-s + ⋯
L(s)  = 1  + (−0.670 − 0.387i)2-s + (−0.137 − 0.990i)3-s + (−0.199 − 0.346i)4-s − 0.447·5-s + (−0.291 + 0.717i)6-s + (0.522 − 0.852i)7-s + 1.08i·8-s + (−0.962 + 0.272i)9-s + (0.300 + 0.173i)10-s − 1.78i·11-s + (−0.315 + 0.245i)12-s + (−0.250 − 0.144i)13-s + (−0.680 + 0.369i)14-s + (0.0614 + 0.442i)15-s + (0.220 − 0.381i)16-s + (−0.778 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.879 - 0.476i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.879 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120621 + 0.475551i\)
\(L(\frac12)\) \(\approx\) \(0.120621 + 0.475551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.237 + 1.71i)T \)
5 \( 1 + T \)
7 \( 1 + (-1.38 + 2.25i)T \)
good2 \( 1 + (0.948 + 0.547i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 5.93iT - 11T^{2} \)
13 \( 1 + (0.902 + 0.521i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.21 - 5.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.377 + 0.217i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.09iT - 23T^{2} \)
29 \( 1 + (5.13 - 2.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.68 + 2.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.58 + 2.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.48 + 6.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.30 + 7.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0462 + 0.0800i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.34 + 3.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.06 + 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.684i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.48 + 2.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (-11.4 - 6.63i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.413 + 0.716i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.40 + 9.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.579 + 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.320 + 0.184i)T + (48.5 - 84.0i)T^{2} \)
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

−11.05319507177195280212391829796, −10.54829729509256081422353337968, −9.004400115202016090148855105669, −8.295507570181004627749111556173, −7.56780805991914441370266418826, −6.24434304339424024814044995159, −5.26680144303240675625959533948, −3.59236776474557540163866344331, −1.77824510062966801217240561570, −0.45157182514379458826118010727, 2.68849446022150079407280190529, 4.37632680156457055683107452948, 4.87086869091641939522643059794, 6.58049852699815559441185988919, 7.62801046620558700804592909144, 8.556726504384634971861502261913, 9.416598191538094356050901486944, 9.926550531253988286526867898883, 11.28573673337621648843511376938, 12.02000621911792965949602717986

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