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Properties

Label 2-3332-3332.2991-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.801 + 0.598i$
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  + (−0.955 − 0.294i)2-s + (0.680 − 1.73i)3-s + (0.826 + 0.563i)4-s + (−1.16 + 1.45i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−1.80 − 1.67i)9-s + (0.432 − 0.400i)11-s + (1.53 − 1.04i)12-s + (0.425 + 1.86i)13-s + (−0.930 + 0.365i)14-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (1.23 + 2.13i)18-s + (−0.548 − 1.77i)21-s + (−0.531 + 0.255i)22-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.680 − 1.73i)3-s + (0.826 + 0.563i)4-s + (−1.16 + 1.45i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−1.80 − 1.67i)9-s + (0.432 − 0.400i)11-s + (1.53 − 1.04i)12-s + (0.425 + 1.86i)13-s + (−0.930 + 0.365i)14-s + (0.365 + 0.930i)16-s + (0.0747 + 0.997i)17-s + (1.23 + 2.13i)18-s + (−0.548 − 1.77i)21-s + (−0.531 + 0.255i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129785126\)
\(L(\frac12)\) \(\approx\) \(1.129785126\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
good3 \( 1 + (-0.680 + 1.73i)T + (-0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.432 + 0.400i)T + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (-0.425 - 1.86i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.145 + 1.94i)T + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.781 + 1.35i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.268 - 0.129i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.680 - 1.17i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.535 - 0.496i)T + (0.0747 + 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.596760137954628440600572882078, −7.896939969046150724403692092551, −7.13104975852026777013565685220, −6.56956408363180701341923435470, −6.15533549906213736285507990164, −4.35804365274743997184255202044, −3.50837463558891699031874525459, −2.30045149430903191123798031708, −1.76019175476058300336264675460, −0.901598226761024685228963230808, 1.57287402742528229654710940885, 2.95912570345462702814190616330, 3.26835949037716734306000067800, 4.71505613279458848519433722477, 5.31203645735981652546298969982, 5.78906826701536992621465256367, 7.27626404442496916857092439188, 7.889983556033266645579080378745, 8.555717974395225026696154433869, 9.198437556957560837366248632461

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