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Properties

Label 2-3332-476.179-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.591 + 0.806i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (1.83 − 0.241i)5-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (−1.83 − 0.241i)10-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.499 + 0.866i)18-s + (1.70 + 0.707i)20-s + (2.33 − 0.624i)25-s + (−0.707 + 1.70i)29-s + (−0.258 − 0.965i)32-s + i·34-s + (0.707 − 0.707i)36-s + (−0.241 − 1.83i)37-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (1.83 − 0.241i)5-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (−1.83 − 0.241i)10-s + (0.500 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.499 + 0.866i)18-s + (1.70 + 0.707i)20-s + (2.33 − 0.624i)25-s + (−0.707 + 1.70i)29-s + (−0.258 − 0.965i)32-s + i·34-s + (0.707 − 0.707i)36-s + (−0.241 − 1.83i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.591 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178797641\)
\(L(\frac12)\) \(\approx\) \(1.178797641\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 \)
17 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + (-1.83 + 0.241i)T + (0.965 - 0.258i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T^{2} \)
29 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)T^{2} \)
41 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.465 + 0.607i)T + (-0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-1.12 - 1.46i)T + (-0.258 + 0.965i)T^{2} \)
79 \( 1 + (-0.258 - 0.965i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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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

−9.110828298830984191080460170963, −8.190882151226081897152559878179, −6.94129362154883172385819444913, −6.76673983150122782201989378344, −5.77634127640951090814086366346, −5.15021054873857886869126339166, −3.76178856446688160864478797329, −2.76118534028992654072599632322, −1.93948936228206107489917128871, −1.02476192532023227292430811050, 1.51153526522633743432312654890, 2.08689144414420815686726409243, 2.89729972599161214552024725684, 4.50760770260215137314225282240, 5.52899994415112540416601419356, 5.98219307137098662072714620911, 6.65987650379886392734646452299, 7.47503131711983803137173913595, 8.287443363111742291745135987541, 8.939936003397893956228016268351

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