↑

Properties

Label 2-315-15.2-c1-0-3
Degree $2$
Conductor $315$
Sign $0.802 - 0.596i$
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  + (−1.28 + 1.28i)2-s − 1.28i·4-s + (−0.565 − 2.16i)5-s + (0.707 + 0.707i)7-s + (−0.918 − 0.918i)8-s + (3.49 + 2.04i)10-s + 4.14i·11-s + (4.57 − 4.57i)13-s − 1.81·14-s + 4.92·16-s + (5.27 − 5.27i)17-s + 3.06i·19-s + (−2.77 + 0.725i)20-s + (−5.30 − 5.30i)22-s + (3.82 + 3.82i)23-s + ⋯
L(s)  = 1  + (−0.905 + 0.905i)2-s − 0.641i·4-s + (−0.252 − 0.967i)5-s + (0.267 + 0.267i)7-s + (−0.324 − 0.324i)8-s + (1.10 + 0.647i)10-s + 1.24i·11-s + (1.26 − 1.26i)13-s − 0.484·14-s + 1.23·16-s + (1.27 − 1.27i)17-s + 0.702i·19-s + (−0.620 + 0.162i)20-s + (−1.13 − 1.13i)22-s + (0.797 + 0.797i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.802 - 0.596i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795195 + 0.263263i\)
\(L(\frac12)\) \(\approx\) \(0.795195 + 0.263263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.565 + 2.16i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.28 - 1.28i)T - 2iT^{2} \)
11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \)
17 \( 1 + (-5.27 + 5.27i)T - 17iT^{2} \)
19 \( 1 - 3.06iT - 19T^{2} \)
23 \( 1 + (-3.82 - 3.82i)T + 23iT^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + 2.42T + 31T^{2} \)
37 \( 1 + (0.834 + 0.834i)T + 37iT^{2} \)
41 \( 1 + 4.19iT - 41T^{2} \)
43 \( 1 + (-3.71 + 3.71i)T - 43iT^{2} \)
47 \( 1 + (3.14 - 3.14i)T - 47iT^{2} \)
53 \( 1 + (-4.79 - 4.79i)T + 53iT^{2} \)
59 \( 1 - 1.97T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 + (10.7 + 10.7i)T + 67iT^{2} \)
71 \( 1 + 9.76iT - 71T^{2} \)
73 \( 1 + (-0.978 + 0.978i)T - 73iT^{2} \)
79 \( 1 + 4.09iT - 79T^{2} \)
83 \( 1 + (3.68 + 3.68i)T + 83iT^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 + (3.07 + 3.07i)T + 97iT^{2} \)
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

−11.97693573147594786791371911965, −10.49171586661188384886162135216, −9.539687450962233892924037918714, −8.824966855521925325271417390534, −7.86829378252323760535175666027, −7.38734509713773695166704341621, −5.89300548631258319663258208105, −5.05717439545029472578019679286, −3.45095396170603426559265730050, −1.08087976811984294688981498893, 1.25415556064352565915564887791, 2.88171366013468667763269588187, 3.89263552595831133750671769977, 5.83485231981957918007263356163, 6.76354278571626272748719108978, 8.221034056495599892732326429027, 8.707680312823022390204289747705, 9.942209270714882073928349942356, 10.82621258900157695893432684227, 11.17068650084379431656242754836

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