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Properties

Label 2-1690-13.3-c1-0-13
Degree $2$
Conductor $1690$
Sign $0.189 - 0.981i$
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 + (1.36 + 2.36i)3-s + (−0.499 + 0.866i)4-s − 5-s + (1.36 − 2.36i)6-s + (1.5 − 2.59i)7-s + 0.999·8-s + (−2.23 + 3.86i)9-s + (0.5 + 0.866i)10-s + (−1.5 − 2.59i)11-s − 2.73·12-s − 3·14-s + (−1.36 − 2.36i)15-s + (−0.5 − 0.866i)16-s + (−1.09 + 1.90i)17-s + 4.46·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.788 + 1.36i)3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.557 − 0.965i)6-s + (0.566 − 0.981i)7-s + 0.353·8-s + (−0.744 + 1.28i)9-s + (0.158 + 0.273i)10-s + (−0.452 − 0.783i)11-s − 0.788·12-s − 0.801·14-s + (−0.352 − 0.610i)15-s + (−0.125 − 0.216i)16-s + (−0.266 + 0.461i)17-s + 1.05·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490197628\)
\(L(\frac12)\) \(\approx\) \(1.490197628\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.09 - 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.23 - 5.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.26 - 2.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 - 8.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (-5.59 - 9.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.19 - 9i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.66T + 73T^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 + (8.59 + 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.56 + 13.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.676609713944382425378664504212, −8.689045190876568669584285493418, −8.225332992281142454162580741342, −7.64435119581789676432544994723, −6.29254301033284458058803726679, −4.88725846257365959316905017923, −4.35104266253413713642644959802, −3.53208252165982628395605997912, −2.90229547069996658656909013299, −1.33327784754200307017190985252, 0.61440805282310129172825457366, 2.21776201677307074824973015224, 2.53914795657387758308457569853, 4.30457352042420261228206353138, 5.19132636511584855336809918458, 6.31978851385086185502573350421, 6.93394612366929743507222958328, 7.75245859325134347619495870892, 8.141462829588489432557501505974, 8.934258822752726654903639086801

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