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Properties

Label 2-3380-3380.2983-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.998 - 0.0598i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  + (−0.0804 − 0.996i)2-s + (−0.987 + 0.160i)4-s + (0.278 + 0.960i)5-s + (0.239 + 0.970i)8-s + (0.903 + 0.428i)9-s + (0.935 − 0.354i)10-s + (−0.692 + 0.721i)13-s + (0.948 − 0.316i)16-s + (0.0326 − 1.62i)17-s + (0.354 − 0.935i)18-s + (−0.428 − 0.903i)20-s + (−0.845 + 0.534i)25-s + (0.774 + 0.632i)26-s + (−0.00648 − 0.0802i)29-s + (−0.391 − 0.919i)32-s + ⋯
L(s)  = 1  + (−0.0804 − 0.996i)2-s + (−0.987 + 0.160i)4-s + (0.278 + 0.960i)5-s + (0.239 + 0.970i)8-s + (0.903 + 0.428i)9-s + (0.935 − 0.354i)10-s + (−0.692 + 0.721i)13-s + (0.948 − 0.316i)16-s + (0.0326 − 1.62i)17-s + (0.354 − 0.935i)18-s + (−0.428 − 0.903i)20-s + (−0.845 + 0.534i)25-s + (0.774 + 0.632i)26-s + (−0.00648 − 0.0802i)29-s + (−0.391 − 0.919i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.998 - 0.0598i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2983, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.998 - 0.0598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146048589\)
\(L(\frac12)\) \(\approx\) \(1.146048589\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0804 + 0.996i)T \)
5 \( 1 + (-0.278 - 0.960i)T \)
13 \( 1 + (0.692 - 0.721i)T \)
good3 \( 1 + (-0.903 - 0.428i)T^{2} \)
7 \( 1 + (0.0402 - 0.999i)T^{2} \)
11 \( 1 + (0.774 + 0.632i)T^{2} \)
17 \( 1 + (-0.0326 + 1.62i)T + (-0.999 - 0.0402i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.00648 + 0.0802i)T + (-0.987 + 0.160i)T^{2} \)
31 \( 1 + (-0.239 - 0.970i)T^{2} \)
37 \( 1 + (-1.80 - 0.768i)T + (0.692 + 0.721i)T^{2} \)
41 \( 1 + (0.334 - 1.48i)T + (-0.903 - 0.428i)T^{2} \)
43 \( 1 + (0.721 + 0.692i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
53 \( 1 + (-1.71 - 1.03i)T + (0.464 + 0.885i)T^{2} \)
59 \( 1 + (-0.600 + 0.799i)T^{2} \)
61 \( 1 + (-0.297 - 1.02i)T + (-0.845 + 0.534i)T^{2} \)
67 \( 1 + (0.948 + 0.316i)T^{2} \)
71 \( 1 + (-0.0804 - 0.996i)T^{2} \)
73 \( 1 + (0.822 + 0.568i)T + (0.354 + 0.935i)T^{2} \)
79 \( 1 + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.568 + 0.822i)T^{2} \)
89 \( 1 + (-0.516 - 1.92i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.94 + 0.397i)T + (0.919 + 0.391i)T^{2} \)
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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

−9.214349730966674848630919289616, −7.998387348552814789396544604999, −7.37871354649674081099134835968, −6.72773629835440602094908168820, −5.63726138626871939466034218023, −4.67751061679624060328102732282, −4.17190874394150654148736430141, −2.88789438938714525410705739853, −2.48411557592939922242287064975, −1.32444168758205521953839142876, 0.795826740535583679021556653859, 1.99328173426206640613762379127, 3.71561121091759525473490551670, 4.24911957174029951147404190985, 5.16923031143953852018133052153, 5.75614005416195672158009551595, 6.52380005813270831340528363289, 7.36727315569603708123936676087, 8.025188553210630181581481392681, 8.675540180871886681937538453285

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