↑

Properties

Label 2-2513-2513.188-c0-0-0
Degree $2$
Conductor $2513$
Sign $0.927 + 0.374i$
Analytic cond. $1.25415$
Root an. cond. $1.11988$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 0.0815i)2-s + (1.39 − 0.147i)4-s + (0.997 − 0.0701i)7-s + (0.613 − 0.0977i)8-s + (0.554 − 0.832i)9-s + (−1.03 − 0.201i)11-s + (1.53 − 0.189i)14-s + (−0.428 + 0.0916i)16-s + (0.790 − 1.33i)18-s + (−1.61 − 0.227i)22-s + (1.32 − 1.29i)23-s + (−0.652 + 0.757i)25-s + (1.37 − 0.244i)28-s + (1.12 + 1.13i)29-s + (−1.25 + 0.338i)32-s + ⋯
L(s)  = 1  + (1.54 − 0.0815i)2-s + (1.39 − 0.147i)4-s + (0.997 − 0.0701i)7-s + (0.613 − 0.0977i)8-s + (0.554 − 0.832i)9-s + (−1.03 − 0.201i)11-s + (1.53 − 0.189i)14-s + (−0.428 + 0.0916i)16-s + (0.790 − 1.33i)18-s + (−1.61 − 0.227i)22-s + (1.32 − 1.29i)23-s + (−0.652 + 0.757i)25-s + (1.37 − 0.244i)28-s + (1.12 + 1.13i)29-s + (−1.25 + 0.338i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2513\)    =    \(7 \cdot 359\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(1.25415\)
Root analytic conductor: \(1.11988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2513} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2513,\ (\ :0),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.960479369\)
\(L(\frac12)\) \(\approx\) \(2.960479369\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.997 + 0.0701i)T \)
359 \( 1 + (-0.950 + 0.310i)T \)
good2 \( 1 + (-1.54 + 0.0815i)T + (0.994 - 0.105i)T^{2} \)
3 \( 1 + (-0.554 + 0.832i)T^{2} \)
5 \( 1 + (0.652 - 0.757i)T^{2} \)
11 \( 1 + (1.03 + 0.201i)T + (0.926 + 0.376i)T^{2} \)
13 \( 1 + (0.873 - 0.487i)T^{2} \)
17 \( 1 + (-0.525 - 0.851i)T^{2} \)
19 \( 1 + (0.998 - 0.0526i)T^{2} \)
23 \( 1 + (-1.32 + 1.29i)T + (0.0263 - 0.999i)T^{2} \)
29 \( 1 + (-1.12 - 1.13i)T + (-0.00877 + 0.999i)T^{2} \)
31 \( 1 + (0.774 + 0.632i)T^{2} \)
37 \( 1 + (1.08 - 1.34i)T + (-0.217 - 0.976i)T^{2} \)
41 \( 1 + (0.539 + 0.841i)T^{2} \)
43 \( 1 + (0.938 + 0.980i)T + (-0.0438 + 0.999i)T^{2} \)
47 \( 1 + (-0.234 - 0.972i)T^{2} \)
53 \( 1 + (1.12 - 0.655i)T + (0.494 - 0.868i)T^{2} \)
59 \( 1 + (0.974 + 0.226i)T^{2} \)
61 \( 1 + (-0.583 - 0.812i)T^{2} \)
67 \( 1 + (-0.109 - 0.491i)T + (-0.905 + 0.424i)T^{2} \)
71 \( 1 + (0.501 - 1.48i)T + (-0.796 - 0.604i)T^{2} \)
73 \( 1 + (0.974 - 0.226i)T^{2} \)
79 \( 1 + (-0.450 - 0.582i)T + (-0.251 + 0.967i)T^{2} \)
83 \( 1 + (0.919 + 0.392i)T^{2} \)
89 \( 1 + (-0.716 + 0.697i)T^{2} \)
97 \( 1 + (0.217 - 0.976i)T^{2} \)
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

−8.862007844214326777339659851883, −8.297033679733821638097604438119, −7.11879587747325107911446337386, −6.71080490426590492374152044784, −5.62593459399497021715411822264, −4.97781824704188564743077128400, −4.46615995874524761390808211030, −3.42623443745505226382064452365, −2.72357295921999687079148423426, −1.45855021313540644032199562382, 1.80649238710530078873822126021, 2.61091024918981788300530002330, 3.65765665135542205468007913143, 4.73860149583413354134501751287, 4.92932078646052951807484898583, 5.71833502014497103690461520584, 6.67596531996299134174052271988, 7.62100893220453864236920417419, 7.967135155338752110077942721688, 9.103628462782444042901513925662

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