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Properties

Label 2-3380-3380.3039-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.505 - 0.862i$
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 + (−0.632 − 0.774i)5-s + (0.354 + 0.935i)8-s + (0.919 + 0.391i)9-s + (0.970 + 0.239i)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 + (−0.919 + 0.391i)20-s + (−0.200 + 0.979i)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 + (−0.632 − 0.774i)5-s + (0.354 + 0.935i)8-s + (0.919 + 0.391i)9-s + (0.970 + 0.239i)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 + (−0.919 + 0.391i)20-s + (−0.200 + 0.979i)25-s + (−0.692 + 0.721i)26-s + (−1.51 + 1.13i)29-s + (0.996 − 0.0804i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.505 - 0.862i$
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: \((2,\ 3380,\ (\ :0),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7903607011\)
\(L(\frac12)\) \(\approx\) \(0.7903607011\)
\(L(1)\) 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 \( 1 + (0.632 + 0.774i)T \)
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

−8.924917027231125103722138769094, −8.050246349571553325422314178672, −7.70810518215222716394718595449, −6.81919181665353932974920673061, −5.99652001824425137901696054325, −5.28904496889671700215165254666, −4.32221305010826784915057495014, −3.67081830322112228123490338371, −1.91549290350245844866887062904, −1.18961919015281909503314320660, 0.72582999335767260654957988643, 2.09085242114672720329849936199, 2.99951522520957762777229296136, 3.87294499914913382449920757129, 4.39439670389017835003490195159, 5.84906608424385594936867315005, 6.85400630240297389531306610021, 7.25270580697126242706092425497, 7.87239716641160172572714775380, 8.817442557404683647281217230001

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