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Properties

Label 2-3332-3332.1359-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  + (0.222 + 0.974i)2-s + (−0.974 − 1.22i)3-s + (−0.900 + 0.433i)4-s + (0.974 − 1.22i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.321 + 1.40i)9-s + (−0.433 − 1.90i)11-s + (1.40 + 0.678i)12-s + (−0.0990 − 0.433i)13-s + (0.781 + 0.623i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 1.44·18-s + (−1.52 − 0.347i)21-s + (1.75 − 0.846i)22-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.974 − 1.22i)3-s + (−0.900 + 0.433i)4-s + (0.974 − 1.22i)6-s + (0.781 − 0.623i)7-s + (−0.623 − 0.781i)8-s + (−0.321 + 1.40i)9-s + (−0.433 − 1.90i)11-s + (1.40 + 0.678i)12-s + (−0.0990 − 0.433i)13-s + (0.781 + 0.623i)14-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 1.44·18-s + (−1.52 − 0.347i)21-s + (1.75 − 0.846i)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: $\chi_{3332} (1359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4273583382\)
\(L(\frac12)\) \(\approx\) \(0.4273583382\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (-0.781 + 0.623i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (0.974 + 1.22i)T + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.75 - 0.846i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 - 1.56T + T^{2} \)
37 \( 1 + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.900 + 0.433i)T^{2} \)
79 \( 1 + 1.94T + T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
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

−8.104545540949305425919616301160, −7.70504125710573675664340503245, −7.01288259300451560212850498626, −6.11543972583117048913994141812, −5.79793878195699069812163193371, −5.03638973199086095671098649627, −4.08303706359951106917795204764, −2.96096198791885684176359945591, −1.40432744376348382392468374215, −0.27448086789833278714479463189, 1.86710255932552172016212034657, 2.50349385747068693840208406740, 4.06081075910682148603794353477, 4.60548817932567424209086391866, 4.76474603490040270418594549273, 5.84886449894446984705956300999, 6.47398277543495059303142276961, 7.87661464792931116646939910901, 8.607750376940455559255080502022, 9.460091179807245214618354741875

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