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Properties

Label 2-3380-13.12-c1-0-17
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  − 0.0947·3-s − i·5-s − 0.826i·7-s − 2.99·9-s + 1.73i·11-s + 0.0947i·15-s − 1.43·17-s + 1.06i·19-s + 0.0783i·21-s − 3.08·23-s − 25-s + 0.567·27-s + 7.45·29-s + 5.84i·31-s − 0.164i·33-s + ⋯
L(s)  = 1  − 0.0547·3-s − 0.447i·5-s − 0.312i·7-s − 0.997·9-s + 0.522i·11-s + 0.0244i·15-s − 0.347·17-s + 0.245i·19-s + 0.0171i·21-s − 0.643·23-s − 0.200·25-s + 0.109·27-s + 1.38·29-s + 1.04i·31-s − 0.0285i·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

\(L(1)\) \(\approx\) \(1.428298475\)
\(L(\frac12)\) \(\approx\) \(1.428298475\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0947T + 3T^{2} \)
7 \( 1 + 0.826iT - 7T^{2} \)
11 \( 1 - 1.73iT - 11T^{2} \)
17 \( 1 + 1.43T + 17T^{2} \)
19 \( 1 - 1.06iT - 19T^{2} \)
23 \( 1 + 3.08T + 23T^{2} \)
29 \( 1 - 7.45T + 29T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 - 0.983iT - 37T^{2} \)
41 \( 1 + 4.26iT - 41T^{2} \)
43 \( 1 - 9.54T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 0.334T + 53T^{2} \)
59 \( 1 - 11.5iT - 59T^{2} \)
61 \( 1 - 2.71T + 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 0.252T + 79T^{2} \)
83 \( 1 - 5.67iT - 83T^{2} \)
89 \( 1 - 4.59iT - 89T^{2} \)
97 \( 1 + 9.53iT - 97T^{2} \)
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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

−8.684171471491328488395489766266, −7.972332597723095953986475589717, −7.20222207658697134349230030594, −6.33733453114971975882530829556, −5.64019697300556025237972940995, −4.79463987124786041052747031498, −4.08513420701201320816498788324, −3.03584033439023002778487726221, −2.09906875004037978981109085270, −0.816829425341699260314534919163, 0.59516300462321637334736266986, 2.23353499342802024713282035244, 2.87100313799707752753737409429, 3.82235474113957327846158516488, 4.78008991258862861431691273306, 5.78757913399381729395698678397, 6.15603271755888049251567965658, 7.05403505046409445337089134079, 7.960194457322392265587749724545, 8.549297414373189541712116451881

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