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Properties

Label 2-3332-3332.2787-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.518 + 0.855i$
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  + (0.623 − 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−1.62 + 0.781i)6-s + (0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (1.40 + 1.75i)9-s + (−0.777 + 0.974i)11-s + (−0.400 + 1.75i)12-s + (0.777 − 0.974i)13-s + (0.900 − 0.433i)14-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 2.24·18-s + (−1.12 − 1.40i)21-s + (0.277 + 1.21i)22-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−1.62 + 0.781i)6-s + (0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (1.40 + 1.75i)9-s + (−0.777 + 0.974i)11-s + (−0.400 + 1.75i)12-s + (0.777 − 0.974i)13-s + (0.900 − 0.433i)14-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 2.24·18-s + (−1.12 − 1.40i)21-s + (0.277 + 1.21i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.518 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021723338\)
\(L(\frac12)\) \(\approx\) \(1.021723338\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 + 1.24T + T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
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

−8.620002589089613447096956715568, −7.76570939159396396936076542064, −7.04189358124036685068515440408, −5.99553490347317809545188594790, −5.68359267844200387911740295886, −4.92891450562438377191137973868, −4.37667816082238214350833080187, −2.95024773083053095708513365951, −1.79600152205025435696697965688, −1.15505477340255164107469844123, 0.74815926097795494371333911863, 2.75537476225048687331908415653, 4.01952632883530179143597315265, 4.54704064774992225096683047210, 5.09839701099981106742268361505, 5.79706560646134139941565263017, 6.60343976259870295927130038323, 6.96542793320295559174796891653, 8.262490520590866049362649168920, 8.629857139139583053639834499056

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