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Properties

Label 2-3380-260.159-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.379 - 0.925i$
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  + (−0.5 − 0.866i)2-s + (−0.222 − 0.385i)3-s + (−0.499 + 0.866i)4-s − 5-s + (−0.222 + 0.385i)6-s + (−0.623 + 1.07i)7-s + 0.999·8-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)10-s + 0.445·12-s + 1.24·14-s + (0.222 + 0.385i)15-s + (−0.5 − 0.866i)16-s − 0.801·18-s + (0.499 − 0.866i)20-s + 0.554·21-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.222 − 0.385i)3-s + (−0.499 + 0.866i)4-s − 5-s + (−0.222 + 0.385i)6-s + (−0.623 + 1.07i)7-s + 0.999·8-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)10-s + 0.445·12-s + 1.24·14-s + (0.222 + 0.385i)15-s + (−0.5 − 0.866i)16-s − 0.801·18-s + (0.499 − 0.866i)20-s + 0.554·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2538149746\)
\(L(\frac12)\) \(\approx\) \(0.2538149746\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
89 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.885860786711083274981955878724, −8.351584441261844183347602623792, −7.66293645022237291130290649131, −6.71629215947875755728918631402, −6.17751035484244575357416587632, −4.86409512510822916365026245168, −4.11807476793373432206960243010, −3.22111275934272728147375506813, −2.52836156525114322046863923973, −1.21179371820763238897301049975, 0.20745998200690397890445546194, 1.71038263111150294552369137994, 3.52363868234909368847385501351, 4.07388842598320703738963717183, 4.87389950817400574692264746501, 5.64515960693795450875188894823, 6.71243007939031383066406470283, 7.21417828845561498054966838426, 7.85738540773953637506278437524, 8.364268395249710326821783786843

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