Properties

Label 2-690-5.2-c2-0-12
Degree $2$
Conductor $690$
Sign $0.313 - 0.949i$
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 + (−0.435 + 4.98i)5-s − 2.44·6-s + (6.78 + 6.78i)7-s + (2 − 2i)8-s − 2.99i·9-s + (5.41 − 4.54i)10-s + 9.68·11-s + (2.44 + 2.44i)12-s + (−12.5 + 12.5i)13-s − 13.5i·14-s + (5.56 + 6.63i)15-s − 4·16-s + (−2.02 − 2.02i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.0871 + 0.996i)5-s − 0.408·6-s + (0.969 + 0.969i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.541 − 0.454i)10-s + 0.880·11-s + (0.204 + 0.204i)12-s + (−0.965 + 0.965i)13-s − 0.969i·14-s + (0.371 + 0.442i)15-s − 0.250·16-s + (−0.119 − 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.313 - 0.949i$
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: \((2,\ 690,\ (\ :1),\ 0.313 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.465858384\)
\(L(\frac12)\) \(\approx\) \(1.465858384\)
\(L(2)\) 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 \( 1 + (0.435 - 4.98i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (-6.78 - 6.78i)T + 49iT^{2} \)
11 \( 1 - 9.68T + 121T^{2} \)
13 \( 1 + (12.5 - 12.5i)T - 169iT^{2} \)
17 \( 1 + (2.02 + 2.02i)T + 289iT^{2} \)
19 \( 1 - 5.79iT - 361T^{2} \)
29 \( 1 - 25.2iT - 841T^{2} \)
31 \( 1 + 15.3T + 961T^{2} \)
37 \( 1 + (25.5 + 25.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 31.9T + 1.68e3T^{2} \)
43 \( 1 + (15.7 - 15.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-52.4 - 52.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (38.7 - 38.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 9.40iT - 3.48e3T^{2} \)
61 \( 1 + 109.T + 3.72e3T^{2} \)
67 \( 1 + (-67.8 - 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 112.T + 5.04e3T^{2} \)
73 \( 1 + (-44.3 + 44.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 25.9iT - 6.24e3T^{2} \)
83 \( 1 + (-101. + 101. i)T - 6.88e3iT^{2} \)
89 \( 1 + 43.8iT - 7.92e3T^{2} \)
97 \( 1 + (-88.0 - 88.0i)T + 9.40e3iT^{2} \)
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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

−10.52210210768609303316901435724, −9.344743400924781059426170335997, −8.929425926193100810717073407178, −7.84297875799002141802814324410, −7.13934962479414972738942421158, −6.24344366992063591554289713913, −4.83003225640231380889208106871, −3.57773082000378903275564073917, −2.41560807022884988482277738475, −1.67981298405824100276007437738, 0.58804164487083887607697104792, 1.84402401522144560107363938738, 3.72820466673873089418949126976, 4.72313717327438907365282352027, 5.33314566133773362201457178392, 6.79902389479563335394403352427, 7.77368867100232303367376338620, 8.255410310068294951059830769983, 9.178467948416680162054953451481, 9.901463205134393551062614187491

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