Properties

Label 2-690-115.22-c1-0-11
Degree $2$
Conductor $690$
Sign $0.996 - 0.0864i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.98 + 1.03i)5-s + 1.00·6-s + (−0.124 + 0.124i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−0.666 − 2.13i)10-s − 0.552i·11-s + (−0.707 − 0.707i)12-s + (2.68 − 2.68i)13-s + 0.176·14-s + (−2.13 + 0.666i)15-s − 1.00·16-s + (3.61 − 3.61i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.885 + 0.464i)5-s + 0.408·6-s + (−0.0471 + 0.0471i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.210 − 0.674i)10-s − 0.166i·11-s + (−0.204 − 0.204i)12-s + (0.745 − 0.745i)13-s + 0.0471·14-s + (−0.551 + 0.172i)15-s − 0.250·16-s + (0.877 − 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.996 - 0.0864i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.996 - 0.0864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23619 + 0.0535140i\)
\(L(\frac12)\) \(\approx\) \(1.23619 + 0.0535140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.98 - 1.03i)T \)
23 \( 1 + (-0.0741 - 4.79i)T \)
good7 \( 1 + (0.124 - 0.124i)T - 7iT^{2} \)
11 \( 1 + 0.552iT - 11T^{2} \)
13 \( 1 + (-2.68 + 2.68i)T - 13iT^{2} \)
17 \( 1 + (-3.61 + 3.61i)T - 17iT^{2} \)
19 \( 1 + 1.08T + 19T^{2} \)
29 \( 1 - 8.68iT - 29T^{2} \)
31 \( 1 - 2.45T + 31T^{2} \)
37 \( 1 + (-7.79 + 7.79i)T - 37iT^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 + (1.06 + 1.06i)T + 43iT^{2} \)
47 \( 1 + (0.597 + 0.597i)T + 47iT^{2} \)
53 \( 1 + (-5.95 - 5.95i)T + 53iT^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + 9.12iT - 61T^{2} \)
67 \( 1 + (-5.38 + 5.38i)T - 67iT^{2} \)
71 \( 1 - 8.16T + 71T^{2} \)
73 \( 1 + (-3.12 + 3.12i)T - 73iT^{2} \)
79 \( 1 - 6.76T + 79T^{2} \)
83 \( 1 + (-0.190 - 0.190i)T + 83iT^{2} \)
89 \( 1 + 8.76T + 89T^{2} \)
97 \( 1 + (-1.62 + 1.62i)T - 97iT^{2} \)
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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

−10.63339534621538677051489690280, −9.623705092126370179516192009724, −9.160902676246227224197785766919, −7.970824361064569457232221489509, −6.98109151202465120250312815605, −5.90281853387863431833454175646, −5.18341038917133214455527206532, −3.64879506049978797233936641079, −2.75241062451951935030943372502, −1.17820564114865010394000691743, 1.06537805615339039295369351699, 2.24149487128102952725752438828, 4.18854979889810165694683681376, 5.29973735091999394164639840144, 6.25628969597017133316805172071, 6.63023064798783298760893170727, 8.059708971308967079318928751741, 8.528724023849391133917186693832, 9.709654514024311013755381203610, 10.16659046971514352881783684236

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