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Properties

Label 2-3380-3380.3039-c0-0-1
Degree $2$
Conductor $3380$
Sign $0.272 + 0.962i$
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  + (0.799 − 0.600i)2-s + (0.278 − 0.960i)4-s + (0.996 + 0.0804i)5-s + (−0.354 − 0.935i)8-s + (0.919 + 0.391i)9-s + (0.845 − 0.534i)10-s + (−0.987 + 0.160i)13-s + (−0.845 − 0.534i)16-s + (0.308 − 1.89i)17-s + (0.970 − 0.239i)18-s + (0.354 − 0.935i)20-s + (0.987 + 0.160i)25-s + (−0.692 + 0.721i)26-s + (−1.51 + 1.13i)29-s + (−0.996 + 0.0804i)32-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + (0.278 − 0.960i)4-s + (0.996 + 0.0804i)5-s + (−0.354 − 0.935i)8-s + (0.919 + 0.391i)9-s + (0.845 − 0.534i)10-s + (−0.987 + 0.160i)13-s + (−0.845 − 0.534i)16-s + (0.308 − 1.89i)17-s + (0.970 − 0.239i)18-s + (0.354 − 0.935i)20-s + (0.987 + 0.160i)25-s + (−0.692 + 0.721i)26-s + (−1.51 + 1.13i)29-s + (−0.996 + 0.0804i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.415831909\)
\(L(\frac12)\) \(\approx\) \(2.415831909\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.799 + 0.600i)T \)
5 \( 1 + (-0.996 - 0.0804i)T \)
13 \( 1 + (0.987 - 0.160i)T \)
good3 \( 1 + (-0.919 - 0.391i)T^{2} \)
7 \( 1 + (-0.948 - 0.316i)T^{2} \)
11 \( 1 + (0.692 - 0.721i)T^{2} \)
17 \( 1 + (-0.308 + 1.89i)T + (-0.948 - 0.316i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.51 - 1.13i)T + (0.278 - 0.960i)T^{2} \)
31 \( 1 + (-0.354 - 0.935i)T^{2} \)
37 \( 1 + (-0.0802 - 0.00648i)T + (0.987 + 0.160i)T^{2} \)
41 \( 1 + (-1.17 - 0.240i)T + (0.919 + 0.391i)T^{2} \)
43 \( 1 + (0.987 - 0.160i)T^{2} \)
47 \( 1 + (-0.885 - 0.464i)T^{2} \)
53 \( 1 + (0.299 - 0.113i)T + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.428 - 0.903i)T^{2} \)
61 \( 1 + (0.253 + 0.309i)T + (-0.200 + 0.979i)T^{2} \)
67 \( 1 + (0.845 - 0.534i)T^{2} \)
71 \( 1 + (0.799 - 0.600i)T^{2} \)
73 \( 1 + (0.120 - 0.992i)T + (-0.970 - 0.239i)T^{2} \)
79 \( 1 + (-0.885 - 0.464i)T^{2} \)
83 \( 1 + (-0.120 + 0.992i)T^{2} \)
89 \( 1 + (-0.804 + 0.464i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.0457 + 1.13i)T + (-0.996 - 0.0804i)T^{2} \)
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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

−9.163926755811699138928247512151, −7.46538019348231829664292148254, −7.17083057481087194283702603547, −6.26064790590975641208094800194, −5.23253485227149957130745962864, −5.02289450042914136213526816940, −4.01878628445207542849945745623, −2.86481741635989871312296551167, −2.24664433399890153799557410673, −1.23344519832702671411379912524, 1.68753402561781008799901291544, 2.51206508107747064574351620582, 3.73504875124385531375264628180, 4.32247514046651495773155658162, 5.30581817745274968620222509630, 5.93859576678052423958021910379, 6.51056833863124135075887623250, 7.39245601642506088711504856330, 7.932706709508652084557109335583, 8.937303006902402586185459554005

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