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Properties

Label 2-690-5.2-c2-0-11
Degree $2$
Conductor $690$
Sign $-0.167 - 0.985i$
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 + (−1.94 − 4.60i)5-s − 2.44·6-s + (−2.18 − 2.18i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (2.65 − 6.55i)10-s + 0.598·11-s + (−2.44 − 2.44i)12-s + (2.44 − 2.44i)13-s − 4.37i·14-s + (8.02 + 3.25i)15-s − 4·16-s + (17.9 + 17.9i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.389 − 0.920i)5-s − 0.408·6-s + (−0.312 − 0.312i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.265 − 0.655i)10-s + 0.0544·11-s + (−0.204 − 0.204i)12-s + (0.188 − 0.188i)13-s − 0.312i·14-s + (0.535 + 0.216i)15-s − 0.250·16-s + (1.05 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.167 - 0.985i$
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: \((2,\ 690,\ (\ :1),\ -0.167 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.597404906\)
\(L(\frac12)\) \(\approx\) \(1.597404906\)
\(L(2)\) 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 \( 1 + (1.94 + 4.60i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (2.18 + 2.18i)T + 49iT^{2} \)
11 \( 1 - 0.598T + 121T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 169iT^{2} \)
17 \( 1 + (-17.9 - 17.9i)T + 289iT^{2} \)
19 \( 1 - 17.4iT - 361T^{2} \)
29 \( 1 - 25.3iT - 841T^{2} \)
31 \( 1 - 49.7T + 961T^{2} \)
37 \( 1 + (-42.3 - 42.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 59.6T + 1.68e3T^{2} \)
43 \( 1 + (37.2 - 37.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-3.72 - 3.72i)T + 2.20e3iT^{2} \)
53 \( 1 + (-66.2 + 66.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 106. iT - 3.48e3T^{2} \)
61 \( 1 - 56.8T + 3.72e3T^{2} \)
67 \( 1 + (5.50 + 5.50i)T + 4.48e3iT^{2} \)
71 \( 1 + 18.4T + 5.04e3T^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - 5.32e3iT^{2} \)
79 \( 1 - 123. iT - 6.24e3T^{2} \)
83 \( 1 + (-34.1 + 34.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 55.0iT - 7.92e3T^{2} \)
97 \( 1 + (-55.4 - 55.4i)T + 9.40e3iT^{2} \)
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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

−10.32966758594859007353271407038, −9.797166496334762541424016136740, −8.445471077417232311754806910601, −8.077757862133631095641196246316, −6.80771052723135508937192046359, −5.87901488185748319242339798138, −5.09515020058024092586142260613, −4.11790494746605341326457196392, −3.35387259163191514543490892807, −1.19540264737127711268884887181, 0.58758612801825933488297318755, 2.35812195907281386383039326458, 3.19768759574167959412074292314, 4.40996217215800364066164409545, 5.53737654265099690604210825401, 6.44490274566657872831766174199, 7.16386845463724254081122191298, 8.160311265172551306738316131599, 9.457522665749432115419733020022, 10.20417374946581713278328888139

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