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Properties

Label 2-3380-3380.3287-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.636 - 0.771i$
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  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (0.992 + 0.120i)8-s + (0.999 + 0.0402i)9-s + (0.822 − 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (−1.86 + 0.264i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (−0.0804 − 0.996i)26-s + (0.297 − 0.470i)29-s + (−0.316 − 0.948i)32-s + ⋯
L(s)  = 1  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (0.992 + 0.120i)8-s + (0.999 + 0.0402i)9-s + (0.822 − 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (−1.86 + 0.264i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (−0.0804 − 0.996i)26-s + (0.297 − 0.470i)29-s + (−0.316 − 0.948i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.636 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5196627163\)
\(L(\frac12)\) \(\approx\) \(0.5196627163\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.534 - 0.845i)T \)
5 \( 1 + (0.919 + 0.391i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
good3 \( 1 + (-0.999 - 0.0402i)T^{2} \)
7 \( 1 + (-0.278 - 0.960i)T^{2} \)
11 \( 1 + (-0.0804 - 0.996i)T^{2} \)
17 \( 1 + (1.86 - 0.264i)T + (0.960 - 0.278i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.297 + 0.470i)T + (-0.428 - 0.903i)T^{2} \)
31 \( 1 + (-0.992 - 0.120i)T^{2} \)
37 \( 1 + (-0.304 - 0.101i)T + (0.799 + 0.600i)T^{2} \)
41 \( 1 + (0.0392 - 1.95i)T + (-0.999 - 0.0402i)T^{2} \)
43 \( 1 + (-0.600 - 0.799i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (-0.173 - 0.221i)T + (-0.239 + 0.970i)T^{2} \)
59 \( 1 + (0.979 + 0.200i)T^{2} \)
61 \( 1 + (-1.32 - 0.565i)T + (0.692 + 0.721i)T^{2} \)
67 \( 1 + (-0.632 - 0.774i)T^{2} \)
71 \( 1 + (-0.534 + 0.845i)T^{2} \)
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 \( 1 + (0.212 - 0.792i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.212 - 1.30i)T + (-0.948 - 0.316i)T^{2} \)
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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

−8.981006371393905168419092948020, −8.206242091617488859676549072401, −7.61526794670532587290278559844, −6.85764956473622315736322195602, −6.45147253622708353457539822182, −5.18829099270649049376682303652, −4.39783030064059146427333948781, −4.15763496687127071099894945484, −2.42067033715399129081140728996, −1.21826353194580871091682354747, 0.40895469539782525920761921555, 1.96772743916898576198132028330, 2.78680572820514859872057900769, 3.80607292742726828764245635577, 4.37499255991878881813139364300, 5.16445292811023289495956592477, 6.79023843460539834981559357321, 7.09551216574757037315525200445, 7.86056146503876331131463831377, 8.620568618769687490255776701305

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