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Properties

Label 2-3380-260.199-c0-0-14
Degree $2$
Conductor $3380$
Sign $-0.820 + 0.571i$
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  + (−0.866 − 0.5i)2-s + (0.623 − 1.07i)3-s + (0.499 + 0.866i)4-s − i·5-s + (−1.07 + 0.623i)6-s + (1.56 − 0.900i)7-s − 0.999i·8-s + (−0.277 − 0.480i)9-s + (−0.5 + 0.866i)10-s + 1.24·12-s − 1.80·14-s + (−1.07 − 0.623i)15-s + (−0.5 + 0.866i)16-s + 0.554i·18-s + (0.866 − 0.499i)20-s − 2.24i·21-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.623 − 1.07i)3-s + (0.499 + 0.866i)4-s − i·5-s + (−1.07 + 0.623i)6-s + (1.56 − 0.900i)7-s − 0.999i·8-s + (−0.277 − 0.480i)9-s + (−0.5 + 0.866i)10-s + 1.24·12-s − 1.80·14-s + (−1.07 − 0.623i)15-s + (−0.5 + 0.866i)16-s + 0.554i·18-s + (0.866 − 0.499i)20-s − 2.24i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266760188\)
\(L(\frac12)\) \(\approx\) \(1.266760188\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - 0.445iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.24iT - T^{2} \)
89 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
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

−8.526499973232757855777133291029, −7.88454536810201912637032837520, −7.35637007287794768827321693541, −6.76376202265842098534768606646, −5.35933508326913723387066209778, −4.51946982499993589330588766097, −3.66052207779540427052490956159, −2.35702418820719239043730764157, −1.56835154410693863282321618764, −1.02753243536270224154717360985, 1.76760095048315719694056695654, 2.51820558738696211067371205816, 3.53882211625115991536429072201, 4.64899961197908974998430039287, 5.31524893277317446342593760752, 6.13366127638711768436287369709, 7.05366350535171270884187937670, 7.904000965319235057048693672424, 8.348308209917263750995728405182, 9.041127581202077932785062683410

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