Properties

Label 2-3380-13.12-c1-0-3
Degree $2$
Conductor $3380$
Sign $-0.960 - 0.277i$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60·3-s i·5-s + 4.33i·7-s − 0.439·9-s − 1.73i·11-s − 1.60i·15-s − 7.50·17-s + 5.37i·19-s + 6.93i·21-s + 1.16·23-s − 25-s − 5.50·27-s − 2.02·29-s − 7.86i·31-s − 2.77i·33-s + ⋯
L(s)  = 1  + 0.923·3-s − 0.447i·5-s + 1.63i·7-s − 0.146·9-s − 0.522i·11-s − 0.413i·15-s − 1.81·17-s + 1.23i·19-s + 1.51i·21-s + 0.242·23-s − 0.200·25-s − 1.05·27-s − 0.375·29-s − 1.41i·31-s − 0.482i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.960 - 0.277i$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3041, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6606893251\)
\(L(\frac12)\) \(\approx\) \(0.6606893251\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 - 1.60T + 3T^{2} \)
7 \( 1 - 4.33iT - 7T^{2} \)
11 \( 1 + 1.73iT - 11T^{2} \)
17 \( 1 + 7.50T + 17T^{2} \)
19 \( 1 - 5.37iT - 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + 2.02T + 29T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 - 9.53iT - 37T^{2} \)
41 \( 1 + 7.73iT - 41T^{2} \)
43 \( 1 + 4.18T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 6.34iT - 59T^{2} \)
61 \( 1 - 3.70T + 61T^{2} \)
67 \( 1 - 5.26iT - 67T^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 - 5.23iT - 73T^{2} \)
79 \( 1 + 8.16T + 79T^{2} \)
83 \( 1 - 0.456iT - 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 2.81iT - 97T^{2} \)
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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

−8.955809598764151079495629627377, −8.325635533988686544052013568448, −7.87479280570231101072792313287, −6.58632359120370621198792180910, −5.90294693352870023708188027104, −5.26021650122930482321911520255, −4.23508782546277127063209257510, −3.28954931493021210100149798914, −2.44336249274355565043184318982, −1.82120365429999100370311423133, 0.15436777573889010360149021476, 1.70166289359425860790954991561, 2.68804658863781236945279392087, 3.46743128771482986917936382382, 4.31497488149998727807518151346, 4.91501226824690967830164758025, 6.34945270081983195862146622860, 6.98348210924875816730900098728, 7.42515609795749630542388376797, 8.270109442446739073688622765892

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