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Properties

Label 2-1690-13.9-c1-0-31
Degree $2$
Conductor $1690$
Sign $0.668 + 0.743i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
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  + (−0.5 + 0.866i)2-s + (0.623 − 1.07i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.623 + 1.07i)6-s + (−0.0990 − 0.171i)7-s + 0.999·8-s + (0.722 + 1.25i)9-s + (0.5 − 0.866i)10-s + (−0.554 + 0.961i)11-s − 1.24·12-s + 0.198·14-s + (−0.623 + 1.07i)15-s + (−0.5 + 0.866i)16-s + (−3.24 − 5.62i)17-s − 1.44·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.359 − 0.623i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.254 + 0.440i)6-s + (−0.0374 − 0.0648i)7-s + 0.353·8-s + (0.240 + 0.417i)9-s + (0.158 − 0.273i)10-s + (−0.167 + 0.289i)11-s − 0.359·12-s + 0.0529·14-s + (−0.160 + 0.278i)15-s + (−0.125 + 0.216i)16-s + (−0.787 − 1.36i)17-s − 0.340·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.668 + 0.743i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.257310709\)
\(L(\frac12)\) \(\approx\) \(1.257310709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + (-0.623 + 1.07i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.0990 + 0.171i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.554 - 0.961i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.24 + 5.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.246 - 0.427i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.38 + 5.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.178 + 0.309i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.27T + 31T^{2} \)
37 \( 1 + (1.10 - 1.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.48 - 7.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.82 + 4.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.43T + 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + (2.30 + 3.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.75 + 4.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 + 3.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.80 + 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.591T + 73T^{2} \)
79 \( 1 - 2.93T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (-8.54 + 14.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.78 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.932413269465556723904441114158, −8.371034334711154314114316720758, −7.60735343521533585643089779237, −6.96417405176777442422643867337, −6.42517216384255038410442718306, −4.97603737123476255211202399234, −4.58739979396699967271776246260, −3.06311432970113898573174204183, −2.04511826045258026137221625670, −0.59460384998647476564785313263, 1.16418898677585104980155273397, 2.57079081216772574238897698699, 3.58931448492922398300321027282, 4.09820461940348858170634575990, 5.12317667928990523874081230076, 6.30717828731049174465106395954, 7.21927394625134912237725865339, 8.169028033315512994527339209523, 8.835881142581052630041844770433, 9.366247080328166012377123417481

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