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Properties

Label 2-3380-3380.903-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.589 + 0.808i$
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  + (−0.999 − 0.0402i)2-s + (0.996 + 0.0804i)4-s + (0.799 − 0.600i)5-s + (−0.992 − 0.120i)8-s + (0.534 − 0.845i)9-s + (−0.822 + 0.568i)10-s + (0.919 + 0.391i)13-s + (0.987 + 0.160i)16-s + (−1.20 − 0.483i)17-s + (−0.568 + 0.822i)18-s + (0.845 − 0.534i)20-s + (0.278 − 0.960i)25-s + (−0.903 − 0.428i)26-s + (1.38 + 0.0557i)29-s + (−0.979 − 0.200i)32-s + ⋯
L(s)  = 1  + (−0.999 − 0.0402i)2-s + (0.996 + 0.0804i)4-s + (0.799 − 0.600i)5-s + (−0.992 − 0.120i)8-s + (0.534 − 0.845i)9-s + (−0.822 + 0.568i)10-s + (0.919 + 0.391i)13-s + (0.987 + 0.160i)16-s + (−1.20 − 0.483i)17-s + (−0.568 + 0.822i)18-s + (0.845 − 0.534i)20-s + (0.278 − 0.960i)25-s + (−0.903 − 0.428i)26-s + (1.38 + 0.0557i)29-s + (−0.979 − 0.200i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.589 + 0.808i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (903, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.589 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010646364\)
\(L(\frac12)\) \(\approx\) \(1.010646364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.999 + 0.0402i)T \)
5 \( 1 + (-0.799 + 0.600i)T \)
13 \( 1 + (-0.919 - 0.391i)T \)
good3 \( 1 + (-0.534 + 0.845i)T^{2} \)
7 \( 1 + (-0.692 + 0.721i)T^{2} \)
11 \( 1 + (-0.903 - 0.428i)T^{2} \)
17 \( 1 + (1.20 + 0.483i)T + (0.721 + 0.692i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1.38 - 0.0557i)T + (0.996 + 0.0804i)T^{2} \)
31 \( 1 + (0.992 + 0.120i)T^{2} \)
37 \( 1 + (-0.309 - 1.51i)T + (-0.919 + 0.391i)T^{2} \)
41 \( 1 + (0.709 + 1.28i)T + (-0.534 + 0.845i)T^{2} \)
43 \( 1 + (0.391 - 0.919i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (-0.587 + 0.460i)T + (0.239 - 0.970i)T^{2} \)
59 \( 1 + (0.316 + 0.948i)T^{2} \)
61 \( 1 + (1.53 - 1.15i)T + (0.278 - 0.960i)T^{2} \)
67 \( 1 + (0.987 - 0.160i)T^{2} \)
71 \( 1 + (-0.999 - 0.0402i)T^{2} \)
73 \( 1 + (-0.464 + 0.885i)T + (-0.568 - 0.822i)T^{2} \)
79 \( 1 + (-0.354 + 0.935i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 + (-0.472 - 1.76i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.02 + 0.838i)T + (0.200 + 0.979i)T^{2} \)
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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

−8.814678501255944745317667450022, −8.280598405347974768028718761410, −7.07645035529050845850393930386, −6.56540677808857560565374631215, −6.00202793846111093446691845982, −4.90922423676498892661127283316, −3.95843043412816852588148850354, −2.82706174600166634732829331031, −1.81304936544492936798242524507, −0.923673467267691046487273913872, 1.35942071324263000592910848995, 2.22207937423604517057563200376, 3.00732106815845347598300849392, 4.23176569932914515224650807994, 5.36255611214540606734691805939, 6.21236266365498116646742623497, 6.66453582977580578335006934080, 7.51324764912870933286023692770, 8.216980772532801274445925035972, 8.912552735382220282906607649759

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