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Properties

Label 2-690-5.3-c2-0-40
Degree $2$
Conductor $690$
Sign $-0.936 + 0.351i$
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 + (4.95 − 0.632i)5-s − 2.44·6-s + (3.11 − 3.11i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (4.32 − 5.59i)10-s − 11.9·11-s + (−2.44 + 2.44i)12-s + (−6.79 − 6.79i)13-s − 6.22i·14-s + (−6.84 − 5.29i)15-s − 4·16-s + (−3.44 + 3.44i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.991 − 0.126i)5-s − 0.408·6-s + (0.444 − 0.444i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.432 − 0.559i)10-s − 1.08·11-s + (−0.204 + 0.204i)12-s + (−0.522 − 0.522i)13-s − 0.444i·14-s + (−0.456 − 0.353i)15-s − 0.250·16-s + (−0.202 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.936 + 0.351i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.887178303\)
\(L(\frac12)\) \(\approx\) \(1.887178303\)
\(L(2)\) 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 \( 1 + (-4.95 + 0.632i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (-3.11 + 3.11i)T - 49iT^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 + (6.79 + 6.79i)T + 169iT^{2} \)
17 \( 1 + (3.44 - 3.44i)T - 289iT^{2} \)
19 \( 1 + 27.8iT - 361T^{2} \)
29 \( 1 + 26.4iT - 841T^{2} \)
31 \( 1 + 16.0T + 961T^{2} \)
37 \( 1 + (-13.6 + 13.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.3T + 1.68e3T^{2} \)
43 \( 1 + (22.4 + 22.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (35.7 - 35.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (41.6 + 41.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 59.3iT - 3.48e3T^{2} \)
61 \( 1 - 9.79T + 3.72e3T^{2} \)
67 \( 1 + (35.1 - 35.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 28.4T + 5.04e3T^{2} \)
73 \( 1 + (2.27 + 2.27i)T + 5.32e3iT^{2} \)
79 \( 1 + 21.6iT - 6.24e3T^{2} \)
83 \( 1 + (48.0 + 48.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 113. iT - 7.92e3T^{2} \)
97 \( 1 + (-43.1 + 43.1i)T - 9.40e3iT^{2} \)
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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

−10.10173555663596261504255869129, −9.275844814788796041207761679217, −8.047509645655380199632950011469, −7.12494989186784432639245085815, −6.09533353344239766840789889818, −5.22247496466544946554122466285, −4.60136657470196094816020746131, −2.88112962451590983807646488597, −1.98344996791271474542022324897, −0.56275482508573433535854153166, 1.89838269320168605418812694933, 3.09412619986553642586811227689, 4.56189528649878761103909809939, 5.32009768388453833114410840148, 5.94350324773634849644070502970, 6.94099977444772066650933115047, 7.971142669046654403943862794178, 8.941533323940449931825489626632, 9.854556586317704185743666969037, 10.57993264417772615877212466428

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