Properties

Label 2-3380-3380.3127-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.999 - 0.00209i$
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  + (−0.721 − 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.948 + 0.316i)5-s + (0.663 − 0.748i)8-s + (0.960 − 0.278i)9-s + (−0.464 − 0.885i)10-s + (0.200 + 0.979i)13-s + (−0.996 + 0.0804i)16-s + (0.810 + 1.22i)17-s + (−0.885 − 0.464i)18-s + (−0.278 + 0.960i)20-s + (0.799 + 0.600i)25-s + (0.534 − 0.845i)26-s + (−1.32 − 1.27i)29-s + (0.774 + 0.632i)32-s + ⋯
L(s)  = 1  + (−0.721 − 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.948 + 0.316i)5-s + (0.663 − 0.748i)8-s + (0.960 − 0.278i)9-s + (−0.464 − 0.885i)10-s + (0.200 + 0.979i)13-s + (−0.996 + 0.0804i)16-s + (0.810 + 1.22i)17-s + (−0.885 − 0.464i)18-s + (−0.278 + 0.960i)20-s + (0.799 + 0.600i)25-s + (0.534 − 0.845i)26-s + (−1.32 − 1.27i)29-s + (0.774 + 0.632i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.999 - 0.00209i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.999 - 0.00209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167045391\)
\(L(\frac12)\) \(\approx\) \(1.167045391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.721 + 0.692i)T \)
5 \( 1 + (-0.948 - 0.316i)T \)
13 \( 1 + (-0.200 - 0.979i)T \)
good3 \( 1 + (-0.960 + 0.278i)T^{2} \)
7 \( 1 + (0.919 + 0.391i)T^{2} \)
11 \( 1 + (0.534 - 0.845i)T^{2} \)
17 \( 1 + (-0.810 - 1.22i)T + (-0.391 + 0.919i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (1.32 + 1.27i)T + (0.0402 + 0.999i)T^{2} \)
31 \( 1 + (-0.663 + 0.748i)T^{2} \)
37 \( 1 + (-1.14 - 1.39i)T + (-0.200 + 0.979i)T^{2} \)
41 \( 1 + (-0.00565 - 0.0398i)T + (-0.960 + 0.278i)T^{2} \)
43 \( 1 + (-0.979 + 0.200i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
53 \( 1 + (1.66 - 0.100i)T + (0.992 - 0.120i)T^{2} \)
59 \( 1 + (0.160 - 0.987i)T^{2} \)
61 \( 1 + (1.13 + 0.380i)T + (0.799 + 0.600i)T^{2} \)
67 \( 1 + (-0.996 - 0.0804i)T^{2} \)
71 \( 1 + (-0.721 - 0.692i)T^{2} \)
73 \( 1 + (0.239 + 0.970i)T + (-0.885 + 0.464i)T^{2} \)
79 \( 1 + (0.568 - 0.822i)T^{2} \)
83 \( 1 + (0.970 - 0.239i)T^{2} \)
89 \( 1 + (0.348 - 0.0933i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.68 - 0.801i)T + (0.632 + 0.774i)T^{2} \)
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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

−9.095620610812882324449421862479, −8.073155848087212609340187899203, −7.50751027430420200686890153963, −6.50033572571692863232239422519, −6.12843499834490827674415959578, −4.75518893150090996659894662727, −3.95009951470118473393600760615, −3.12074236839175373035123119577, −1.91795180034273817603089394560, −1.43515185684285544918304079048, 1.01057786536729681000464750446, 1.90937445054671979582447312033, 3.07877571039229701793632858852, 4.50661448605289618203621065878, 5.27480466635483608425574693247, 5.75391062475750257656612521842, 6.64791846071482246965072900879, 7.47854502585261504197435560351, 7.85380171506592432022089960830, 8.936171906663444152653929177052

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