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Properties

Label 2-1690-13.9-c1-0-8
Degree $2$
Conductor $1690$
Sign $0.522 - 0.852i$
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 − 1.73i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.999 + 1.73i)6-s + (−2 − 3.46i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−3 + 5.19i)11-s − 1.99·12-s + 3.99·14-s + (−1 + 1.73i)15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.408 + 0.707i)6-s + (−0.755 − 1.30i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.904 + 1.56i)11-s − 0.577·12-s + 1.06·14-s + (−0.258 + 0.447i)15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.522 - 0.852i$
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: \((2,\ 1690,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049556284\)
\(L(\frac12)\) \(\approx\) \(1.049556284\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.480363942118803321106283318272, −8.234641683639294565602650658500, −7.70057953488436627275749102365, −7.35016455375421234317084560956, −6.66395483739923710544754496540, −5.64354701366275914632445857865, −4.40827261915677824508079282981, −3.62479847050983447030105644994, −2.23872695508419885099583002876, −1.12403060830478119603931605509, 0.47817888609481973655585691850, 2.74760674215008413100175214019, 2.92667367745940673618242218222, 3.84978199232551646584190063696, 5.03330047009396164755071566366, 5.72770277758386861805479411260, 6.92538210530337525125810061250, 8.101655152059131114303976429913, 8.739893972254374337941201624656, 9.113040490385229011602841007763

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