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Properties

Label 2-612-17.15-c1-0-4
Degree $2$
Conductor $612$
Sign $-0.134 + 0.990i$
Analytic cond. $4.88684$
Root an. cond. $2.21062$
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  + (−2.83 + 1.17i)5-s + (−0.119 − 0.0496i)7-s + (0.738 − 1.78i)11-s − 3.76i·13-s + (−2.73 − 3.08i)17-s + (0.765 + 0.765i)19-s + (1.79 − 4.33i)23-s + (3.10 − 3.10i)25-s + (4.21 − 1.74i)29-s + (−3.70 − 8.93i)31-s + 0.397·35-s + (−3.12 − 7.53i)37-s + (3.47 + 1.44i)41-s + (−7.24 + 7.24i)43-s + 2.96i·47-s + ⋯
L(s)  = 1  + (−1.26 + 0.524i)5-s + (−0.0453 − 0.0187i)7-s + (0.222 − 0.537i)11-s − 1.04i·13-s + (−0.664 − 0.747i)17-s + (0.175 + 0.175i)19-s + (0.374 − 0.904i)23-s + (0.621 − 0.621i)25-s + (0.783 − 0.324i)29-s + (−0.665 − 1.60i)31-s + 0.0672·35-s + (−0.513 − 1.23i)37-s + (0.543 + 0.225i)41-s + (−1.10 + 1.10i)43-s + 0.433i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(4.88684\)
Root analytic conductor: \(2.21062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{612} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 612,\ (\ :1/2),\ -0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492866 - 0.564315i\)
\(L(\frac12)\) \(\approx\) \(0.492866 - 0.564315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + (2.73 + 3.08i)T \)
good5 \( 1 + (2.83 - 1.17i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.119 + 0.0496i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.738 + 1.78i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.76iT - 13T^{2} \)
19 \( 1 + (-0.765 - 0.765i)T + 19iT^{2} \)
23 \( 1 + (-1.79 + 4.33i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-4.21 + 1.74i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3.70 + 8.93i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.12 + 7.53i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.47 - 1.44i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.24 - 7.24i)T - 43iT^{2} \)
47 \( 1 - 2.96iT - 47T^{2} \)
53 \( 1 + (-0.460 - 0.460i)T + 53iT^{2} \)
59 \( 1 + (2.86 - 2.86i)T - 59iT^{2} \)
61 \( 1 + (-11.5 - 4.80i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 + (6.16 + 14.8i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.97 - 2.47i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.719 + 1.73i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.97 - 4.97i)T + 83iT^{2} \)
89 \( 1 - 6.94iT - 89T^{2} \)
97 \( 1 + (-5.12 + 2.12i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.63440321433248026567973414045, −9.535567685292282727538756835005, −8.463399014566948834901479120609, −7.77402692506433471109461565586, −6.96156544724657677860210707594, −5.92528362260278623749255386904, −4.65862966101094183515747542776, −3.64586416141492438454859523003, −2.70242831771907709074437040125, −0.42545423652954962297210226874, 1.59912458303567038817965516285, 3.39608849773137606754362165827, 4.31522138961105513742588106686, 5.11234183318807860693059306933, 6.65389249029701866749240924406, 7.25794928952153453573399751180, 8.412755714298071661748626000418, 8.893537573801895449397987150437, 9.989056066392140283580102948148, 11.04855673226206908701735474911

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