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Properties

Label 2-3380-3380.3167-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.549 - 0.835i$
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  + (0.239 + 0.970i)2-s + (−0.885 + 0.464i)4-s + (−0.748 − 0.663i)5-s + (−0.663 − 0.748i)8-s + (0.239 + 0.970i)9-s + (0.464 − 0.885i)10-s + (0.748 + 0.663i)13-s + (0.568 − 0.822i)16-s + (0.0359 − 0.593i)17-s + (−0.885 + 0.464i)18-s + (0.970 + 0.239i)20-s + (0.120 + 0.992i)25-s + (−0.464 + 0.885i)26-s + (−0.0576 − 0.234i)29-s + (0.935 + 0.354i)32-s + ⋯
L(s)  = 1  + (0.239 + 0.970i)2-s + (−0.885 + 0.464i)4-s + (−0.748 − 0.663i)5-s + (−0.663 − 0.748i)8-s + (0.239 + 0.970i)9-s + (0.464 − 0.885i)10-s + (0.748 + 0.663i)13-s + (0.568 − 0.822i)16-s + (0.0359 − 0.593i)17-s + (−0.885 + 0.464i)18-s + (0.970 + 0.239i)20-s + (0.120 + 0.992i)25-s + (−0.464 + 0.885i)26-s + (−0.0576 − 0.234i)29-s + (0.935 + 0.354i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.549 - 0.835i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026602182\)
\(L(\frac12)\) \(\approx\) \(1.026602182\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.239 - 0.970i)T \)
5 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
good3 \( 1 + (-0.239 - 0.970i)T^{2} \)
7 \( 1 + (-0.120 + 0.992i)T^{2} \)
11 \( 1 + (-0.464 + 0.885i)T^{2} \)
17 \( 1 + (-0.0359 + 0.593i)T + (-0.992 - 0.120i)T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.0576 + 0.234i)T + (-0.885 + 0.464i)T^{2} \)
31 \( 1 + (0.663 + 0.748i)T^{2} \)
37 \( 1 + (-0.583 - 1.53i)T + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (0.638 - 0.814i)T + (-0.239 - 0.970i)T^{2} \)
43 \( 1 + (-0.663 + 0.748i)T^{2} \)
47 \( 1 + (-0.568 - 0.822i)T^{2} \)
53 \( 1 + (0.120 - 1.99i)T + (-0.992 - 0.120i)T^{2} \)
59 \( 1 + (0.935 + 0.354i)T^{2} \)
61 \( 1 + (-1.48 - 1.31i)T + (0.120 + 0.992i)T^{2} \)
67 \( 1 + (0.568 + 0.822i)T^{2} \)
71 \( 1 + (0.239 + 0.970i)T^{2} \)
73 \( 1 + (0.478 - 1.94i)T + (-0.885 - 0.464i)T^{2} \)
79 \( 1 + (0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.970 + 0.239i)T^{2} \)
89 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
97 \( 1 + (1.53 + 1.06i)T + (0.354 + 0.935i)T^{2} \)
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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

−8.606450559888180723353044881188, −8.372550883612376917487392325033, −7.49256724570959727769740991312, −6.98342483860578770450835477970, −6.02518397980310951732323556500, −5.19475341404149921145458732112, −4.53234108516877153339151064366, −3.96928712650664131863567741705, −2.86055256151031234786777190213, −1.26052934773986143875336387974, 0.65256411230447462744789710492, 1.99398749850382576479556778616, 3.18715003243047767085708143274, 3.64907365197819548077669463522, 4.31508124232105559648841157191, 5.46140566234984550666501568650, 6.20767351094194051196386923496, 6.99080294987054879771785548142, 8.005620097171891495263020305371, 8.586250373347428493217256311459

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