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Properties

Label 2-1218-203.104-c1-0-1
Degree $2$
Conductor $1218$
Sign $-0.904 + 0.426i$
Analytic cond. $9.72577$
Root an. cond. $3.11861$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s − 2.17·5-s − 1.00·6-s + (−0.0332 − 2.64i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.53 − 1.53i)10-s + (−3.07 + 3.07i)11-s + (0.707 − 0.707i)12-s + 3.05·13-s + (1.89 + 1.84i)14-s + (−1.53 − 1.53i)15-s − 1.00·16-s + (2.01 + 2.01i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s − 0.973·5-s − 0.408·6-s + (−0.0125 − 0.999i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.486 − 0.486i)10-s + (−0.927 + 0.927i)11-s + (0.204 − 0.204i)12-s + 0.846·13-s + (0.506 + 0.493i)14-s + (−0.397 − 0.397i)15-s − 0.250·16-s + (0.489 + 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1831644825\)
\(L(\frac12)\) \(\approx\) \(0.1831644825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.0332 + 2.64i)T \)
29 \( 1 + (3.12 + 4.38i)T \)
good5 \( 1 + 2.17T + 5T^{2} \)
11 \( 1 + (3.07 - 3.07i)T - 11iT^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 + (-2.01 - 2.01i)T + 17iT^{2} \)
19 \( 1 + (-2.09 - 2.09i)T + 19iT^{2} \)
23 \( 1 + 1.96T + 23T^{2} \)
31 \( 1 + (1.89 + 1.89i)T + 31iT^{2} \)
37 \( 1 + (5.99 + 5.99i)T + 37iT^{2} \)
41 \( 1 + (3.37 - 3.37i)T - 41iT^{2} \)
43 \( 1 + (2.06 - 2.06i)T - 43iT^{2} \)
47 \( 1 + (8.60 - 8.60i)T - 47iT^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + (3.08 + 3.08i)T + 61iT^{2} \)
67 \( 1 - 8.49iT - 67T^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 + (6.46 - 6.46i)T - 73iT^{2} \)
79 \( 1 + (10.6 - 10.6i)T - 79iT^{2} \)
83 \( 1 - 7.41iT - 83T^{2} \)
89 \( 1 + (-12.7 - 12.7i)T + 89iT^{2} \)
97 \( 1 + (-7.25 + 7.25i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00819686459793311310968284561, −9.509141048708827488126927110006, −8.157808587682691541155993467879, −7.926053980074013650990539532884, −7.26111926851783485058589897175, −6.16118161156094094290345472915, −5.01585232030079857322416577347, −4.10119360842413537355802439649, −3.37937717756852430670848906916, −1.67703315454428647354088063434, 0.086880012709424682137016768686, 1.68634748921641369015936181438, 3.09066110819768810952545572398, 3.42134653332049184289683462747, 4.96801256270973337081603473672, 5.91668723839833816063171112822, 7.09668804615041176613891112071, 7.87968583873325150284403838864, 8.540470825215251454819452980171, 8.964135352706912808880595828725

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