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Properties

Label 2-3380-13.12-c1-0-38
Degree $2$
Conductor $3380$
Sign $0.960 + 0.277i$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s + i·5-s − 2.09i·7-s + 4.99·9-s − 1.73i·11-s + 2.82i·15-s + 3.62·17-s + 1.06i·19-s − 5.92i·21-s + 7.81·23-s − 25-s + 5.62·27-s − 0.526·29-s + 5.84i·31-s − 4.89i·33-s + ⋯
L(s)  = 1  + 1.63·3-s + 0.447i·5-s − 0.791i·7-s + 1.66·9-s − 0.522i·11-s + 0.729i·15-s + 0.879·17-s + 0.245i·19-s − 1.29i·21-s + 1.63·23-s − 0.200·25-s + 1.08·27-s − 0.0978·29-s + 1.04i·31-s − 0.852i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3041, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.699070392\)
\(L(\frac12)\) \(\approx\) \(3.699070392\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 2.09iT - 7T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
17 \( 1 - 3.62T + 17T^{2} \)
19 \( 1 - 1.06iT - 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 + 0.526T + 29T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + 9.74iT - 37T^{2} \)
41 \( 1 - 4.26iT - 41T^{2} \)
43 \( 1 + 9.34T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 1.40iT - 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 2.64iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 - 5.52iT - 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−8.715091348934781069750579516741, −7.74722676046300429683887190725, −7.36268908071707774545535317012, −6.65035443080033789390163836760, −5.51005525386630815360759373602, −4.49110552055094303831109209190, −3.38361694125636597608680694803, −3.32590329624150890591192653227, −2.16404612762182486822507371942, −1.05086459116767829827626299732, 1.25651092212177882352233756670, 2.27838989460491709124292579070, 2.96587880813309476487503032959, 3.76761346994668270174851062908, 4.75162944963213471602069850781, 5.45289208787585044548235666346, 6.60104164659693839082029784984, 7.41981365122733859151790730688, 8.052934451283700751563827620505, 8.687952425271133633355699194961

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