↑

Properties

Label 2-315-35.9-c1-0-17
Degree $2$
Conductor $315$
Sign $-0.0667 + 0.997i$
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  + (0.866 + 0.5i)2-s + (−0.500 − 0.866i)4-s + (−1.23 − 1.86i)5-s + (−2.59 − 0.5i)7-s − 3i·8-s + (−0.133 − 2.23i)10-s − 2i·13-s + (−2 − 1.73i)14-s + (0.500 − 0.866i)16-s + (−1.73 + i)17-s + (3 − 5.19i)19-s + (−0.999 + 2i)20-s + (2.59 + 1.5i)23-s + (−1.96 + 4.59i)25-s + (1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.250 − 0.433i)4-s + (−0.550 − 0.834i)5-s + (−0.981 − 0.188i)7-s − 1.06i·8-s + (−0.0423 − 0.705i)10-s − 0.554i·13-s + (−0.534 − 0.462i)14-s + (0.125 − 0.216i)16-s + (−0.420 + 0.242i)17-s + (0.688 − 1.19i)19-s + (−0.223 + 0.447i)20-s + (0.541 + 0.312i)23-s + (−0.392 + 0.919i)25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0667 + 0.997i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786373 - 0.840739i\)
\(L(\frac12)\) \(\approx\) \(0.786373 - 0.840739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.86i)T \)
7 \( 1 + (2.59 + 0.5i)T \)
good2 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-5.19 + 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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

−11.60317753569527490878241552772, −10.38326789203111534216282167866, −9.494097867462170733741034844666, −8.698609719146941821046194015157, −7.35654422661224559653832176368, −6.40879592693807092142269954334, −5.29537782727193847823106833692, −4.43510199009314627191488005456, −3.25469664533210150890835989908, −0.70190913065472863437554892235, 2.66324550789052380189966411942, 3.50495866601066223137159886336, 4.53389543310639782100255310021, 6.00679510082260063121055803455, 7.02764703234062705145594354259, 8.045031713860872722403289987147, 9.123341773678424833492568004072, 10.18754154693846187561978986668, 11.22054768066799051952014240927, 12.02409128828662643020796221794

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