↑

Properties

Label 2-1690-5.4-c1-0-25
Degree $2$
Conductor $1690$
Sign $0.981 - 0.190i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 0.956i·3-s − 4-s + (−0.426 − 2.19i)5-s + 0.956·6-s + 2.15i·7-s + i·8-s + 2.08·9-s + (−2.19 + 0.426i)10-s + 1.30·11-s − 0.956i·12-s + 2.15·14-s + (2.09 − 0.408i)15-s + 16-s + 2.87i·17-s − 2.08i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.552i·3-s − 0.5·4-s + (−0.190 − 0.981i)5-s + 0.390·6-s + 0.813i·7-s + 0.353i·8-s + 0.695·9-s + (−0.694 + 0.134i)10-s + 0.393·11-s − 0.276i·12-s + 0.574·14-s + (0.542 − 0.105i)15-s + 0.250·16-s + 0.697i·17-s − 0.491i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ 0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.578381062\)
\(L(\frac12)\) \(\approx\) \(1.578381062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.426 + 2.19i)T \)
13 \( 1 \)
good3 \( 1 - 0.956iT - 3T^{2} \)
7 \( 1 - 2.15iT - 7T^{2} \)
11 \( 1 - 1.30T + 11T^{2} \)
17 \( 1 - 2.87iT - 17T^{2} \)
19 \( 1 - 2.65T + 19T^{2} \)
23 \( 1 - 5.11iT - 23T^{2} \)
29 \( 1 + 7.96T + 29T^{2} \)
31 \( 1 + 4.12T + 31T^{2} \)
37 \( 1 + 2.98iT - 37T^{2} \)
41 \( 1 - 8.69T + 41T^{2} \)
43 \( 1 - 3.35iT - 43T^{2} \)
47 \( 1 + 7.30iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 - 14.6T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 3.83iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 5.05iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.391529971976940265215107110114, −8.933473008956182202296759877835, −8.015315839120604627522120885384, −7.13553116228403635770739464059, −5.60736776921719724102399233241, −5.33142395509317663819834858399, −4.05651571440962091070893601926, −3.74341624208567890980677370319, −2.19354149987486205671914629237, −1.18679277058021278720196352453, 0.71058893013848685569287001854, 2.20812634981074989451477708301, 3.57298467567553121770716150407, 4.23968067228194930891214984568, 5.40158890715736452656158145340, 6.46325942477859318206997915248, 7.02155657868120272655285688756, 7.45071757163963709596088139876, 8.185262374699413027102737819509, 9.397456191600109003913184085202

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