Properties

Label 2-690-5.2-c2-0-32
Degree $2$
Conductor $690$
Sign $0.980 + 0.195i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (4.99 − 0.177i)5-s + 2.44·6-s + (−4.63 − 4.63i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (5.17 + 4.81i)10-s + 2.27·11-s + (2.44 + 2.44i)12-s + (6.59 − 6.59i)13-s − 9.27i·14-s + (5.90 − 6.33i)15-s − 4·16-s + (16.0 + 16.0i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.999 − 0.0354i)5-s + 0.408·6-s + (−0.662 − 0.662i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.517 + 0.481i)10-s + 0.206·11-s + (0.204 + 0.204i)12-s + (0.507 − 0.507i)13-s − 0.662i·14-s + (0.393 − 0.422i)15-s − 0.250·16-s + (0.946 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.215309417\)
\(L(\frac12)\) \(\approx\) \(3.215309417\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (-4.99 + 0.177i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (4.63 + 4.63i)T + 49iT^{2} \)
11 \( 1 - 2.27T + 121T^{2} \)
13 \( 1 + (-6.59 + 6.59i)T - 169iT^{2} \)
17 \( 1 + (-16.0 - 16.0i)T + 289iT^{2} \)
19 \( 1 + 33.4iT - 361T^{2} \)
29 \( 1 - 8.75iT - 841T^{2} \)
31 \( 1 - 26.0T + 961T^{2} \)
37 \( 1 + (-18.6 - 18.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 4.34T + 1.68e3T^{2} \)
43 \( 1 + (-33.4 + 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (28.7 + 28.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (34.3 - 34.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 71.9iT - 3.48e3T^{2} \)
61 \( 1 + 71.6T + 3.72e3T^{2} \)
67 \( 1 + (80.0 + 80.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 103.T + 5.04e3T^{2} \)
73 \( 1 + (-1.17 + 1.17i)T - 5.32e3iT^{2} \)
79 \( 1 - 134. iT - 6.24e3T^{2} \)
83 \( 1 + (11.8 - 11.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 40.9iT - 7.92e3T^{2} \)
97 \( 1 + (-31.1 - 31.1i)T + 9.40e3iT^{2} \)
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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

−10.16922782751947814790944404747, −9.272656181730805956169392031419, −8.465198957807357226574556520943, −7.42677330632621251860810600694, −6.57891483769406240926013181025, −5.98245446455459346354356537693, −4.84362721774153066325107749072, −3.57582703417283138532783384564, −2.64491731350311212661244582944, −1.04698004881103192869440232366, 1.47659091036342258747391367894, 2.67686847922439919852120460735, 3.51057406021579323694701722616, 4.74832109296280423639256281457, 5.83722580404574220492050288683, 6.30178147382129072853317554471, 7.75931780162269944226792739208, 8.952935963129503576641696907777, 9.642141650357431253503057654288, 10.05165455964656339527638996129

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