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Properties

Label 2-1218-203.104-c1-0-15
Degree $2$
Conductor $1218$
Sign $0.764 + 0.644i$
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 − 1.75·5-s + 1.00·6-s + (−1.92 + 1.81i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.24 − 1.24i)10-s + (−0.974 + 0.974i)11-s + (−0.707 + 0.707i)12-s − 2.81·13-s + (0.0801 − 2.64i)14-s + (1.24 + 1.24i)15-s − 1.00·16-s + (−0.797 − 0.797i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s − 0.784·5-s + 0.408·6-s + (−0.728 + 0.685i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.392 − 0.392i)10-s + (−0.293 + 0.293i)11-s + (−0.204 + 0.204i)12-s − 0.779·13-s + (0.0214 − 0.706i)14-s + (0.320 + 0.320i)15-s − 0.250·16-s + (−0.193 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.764 + 0.644i$
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: \((2,\ 1218,\ (\ :1/2),\ 0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4978850398\)
\(L(\frac12)\) \(\approx\) \(0.4978850398\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 - 1.81i)T \)
29 \( 1 + (-1.61 - 5.13i)T \)
good5 \( 1 + 1.75T + 5T^{2} \)
11 \( 1 + (0.974 - 0.974i)T - 11iT^{2} \)
13 \( 1 + 2.81T + 13T^{2} \)
17 \( 1 + (0.797 + 0.797i)T + 17iT^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \)
23 \( 1 - 8.03T + 23T^{2} \)
31 \( 1 + (7.15 + 7.15i)T + 31iT^{2} \)
37 \( 1 + (0.0852 + 0.0852i)T + 37iT^{2} \)
41 \( 1 + (-2.43 + 2.43i)T - 41iT^{2} \)
43 \( 1 + (-6.66 + 6.66i)T - 43iT^{2} \)
47 \( 1 + (7.23 - 7.23i)T - 47iT^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 2.61iT - 59T^{2} \)
61 \( 1 + (5.33 + 5.33i)T + 61iT^{2} \)
67 \( 1 + 6.79iT - 67T^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (-8.50 + 8.50i)T - 73iT^{2} \)
79 \( 1 + (0.362 - 0.362i)T - 79iT^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + (-6.61 - 6.61i)T + 89iT^{2} \)
97 \( 1 + (-6.73 + 6.73i)T - 97iT^{2} \)
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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

−9.369107016458713936485645374375, −8.942053469815871476534725249769, −7.67017804453779359764148149133, −7.38288611720971891992589550146, −6.44756743734360923440804219543, −5.54115959685633108852183930512, −4.76315973935888549972427450169, −3.37109689875114378224550814428, −2.15026157642791126616554454636, −0.38081180538769276559729558145, 0.826694590610477950223034525684, 2.73637462398184920137093751910, 3.62115601306951769311967394636, 4.46020271846883352385927357181, 5.48109330041608880857334841403, 6.82224265225808192211864503335, 7.31317430283267885923406299028, 8.283642213665629853504032483222, 9.216923983924384563316682052749, 9.867736472643845642391703697232

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