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Properties

Label 2-3377-3377.1841-c0-0-0
Degree $2$
Conductor $3377$
Sign $-0.686 - 0.727i$
Analytic cond. $1.68534$
Root an. cond. $1.29820$
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  + (0.309 + 0.951i)4-s + (0.564 + 1.73i)7-s + (−0.809 − 0.587i)9-s + (−0.104 − 0.994i)11-s + (−0.809 + 0.587i)16-s + (−1.08 + 0.786i)17-s + (−0.604 + 1.86i)19-s + (0.309 − 0.951i)25-s + (−1.47 + 1.07i)28-s + (0.309 − 0.951i)36-s + (−0.0646 − 0.198i)37-s + (−0.309 + 0.951i)41-s + (0.913 − 0.406i)44-s + (−1.89 + 1.37i)49-s + (0.809 + 0.587i)53-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)4-s + (0.564 + 1.73i)7-s + (−0.809 − 0.587i)9-s + (−0.104 − 0.994i)11-s + (−0.809 + 0.587i)16-s + (−1.08 + 0.786i)17-s + (−0.604 + 1.86i)19-s + (0.309 − 0.951i)25-s + (−1.47 + 1.07i)28-s + (0.309 − 0.951i)36-s + (−0.0646 − 0.198i)37-s + (−0.309 + 0.951i)41-s + (0.913 − 0.406i)44-s + (−1.89 + 1.37i)49-s + (0.809 + 0.587i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3377 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3377 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3377\)    =    \(11 \cdot 307\)
Sign: $-0.686 - 0.727i$
Analytic conductor: \(1.68534\)
Root analytic conductor: \(1.29820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3377} (1841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3377,\ (\ :0),\ -0.686 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034172672\)
\(L(\frac12)\) \(\approx\) \(1.034172672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.104 + 0.994i)T \)
307 \( 1 - T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.678893466715032948800706860393, −8.370975033368478925620481557536, −8.047179712230094957687886353395, −6.58899038591649228943071500451, −6.09820689147957687981528648357, −5.51898011703091651562579936930, −4.33051667672874111971719615764, −3.45386907525783546689502289693, −2.64258307864146529760052437244, −1.91980703316231459616870284343, 0.56602291541404048029509275790, 1.86543762897333228799099958594, 2.65125267424124137105933239616, 4.06332887921719577487462673633, 4.92420273715794962177845228450, 5.10945238992810261261558315732, 6.53007693646506782705371395946, 7.06291204409764493123574012901, 7.47039818474821460596451251831, 8.604799439487039111634722272835

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