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Properties

Label 2-690-15.14-c2-0-69
Degree $2$
Conductor $690$
Sign $-0.791 - 0.611i$
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  − 1.41·2-s + (−1.52 − 2.58i)3-s + 2.00·4-s + (−4.64 + 1.85i)5-s + (2.15 + 3.65i)6-s − 10.7i·7-s − 2.82·8-s + (−4.36 + 7.86i)9-s + (6.56 − 2.62i)10-s − 19.8i·11-s + (−3.04 − 5.17i)12-s − 15.8i·13-s + 15.2i·14-s + (11.8 + 9.17i)15-s + 4.00·16-s + 9.43·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.507 − 0.861i)3-s + 0.500·4-s + (−0.928 + 0.371i)5-s + (0.358 + 0.609i)6-s − 1.53i·7-s − 0.353·8-s + (−0.485 + 0.874i)9-s + (0.656 − 0.262i)10-s − 1.80i·11-s + (−0.253 − 0.430i)12-s − 1.21i·13-s + 1.08i·14-s + (0.791 + 0.611i)15-s + 0.250·16-s + 0.555·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.791 - 0.611i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.791 - 0.611i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5771390983\)
\(L(\frac12)\) \(\approx\) \(0.5771390983\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (1.52 + 2.58i)T \)
5 \( 1 + (4.64 - 1.85i)T \)
23 \( 1 + 4.79T \)
good7 \( 1 + 10.7iT - 49T^{2} \)
11 \( 1 + 19.8iT - 121T^{2} \)
13 \( 1 + 15.8iT - 169T^{2} \)
17 \( 1 - 9.43T + 289T^{2} \)
19 \( 1 - 8.14T + 361T^{2} \)
29 \( 1 + 47.6iT - 841T^{2} \)
31 \( 1 + 10.4T + 961T^{2} \)
37 \( 1 + 0.317iT - 1.36e3T^{2} \)
41 \( 1 + 30.0iT - 1.68e3T^{2} \)
43 \( 1 + 13.3iT - 1.84e3T^{2} \)
47 \( 1 - 16.2T + 2.20e3T^{2} \)
53 \( 1 + 77.8T + 2.80e3T^{2} \)
59 \( 1 - 15.6iT - 3.48e3T^{2} \)
61 \( 1 - 12.3T + 3.72e3T^{2} \)
67 \( 1 - 42.8iT - 4.48e3T^{2} \)
71 \( 1 - 97.2iT - 5.04e3T^{2} \)
73 \( 1 - 43.7iT - 5.32e3T^{2} \)
79 \( 1 - 121.T + 6.24e3T^{2} \)
83 \( 1 + 60.3T + 6.88e3T^{2} \)
89 \( 1 - 5.45iT - 7.92e3T^{2} \)
97 \( 1 + 157. iT - 9.40e3T^{2} \)
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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

−10.04822221995320591210640654169, −8.453656748750393576178570001136, −7.86110205647257060830425122438, −7.39285639063263474133472511397, −6.40162621479388160757733572624, −5.50796597625819157935695599088, −3.86173160783415388186826143173, −2.93569908234949653202967729713, −0.906731559618935589532347783960, −0.38229313939143854828715661551, 1.75199664976512701486123585266, 3.23722996281404129780524434645, 4.54757713442383195980460888629, 5.20496277493600391535333984936, 6.43434545751386786752018810762, 7.37338172762837164157142831589, 8.443512754167761133626637857922, 9.358531116438990640199717301455, 9.530953783544369038502944253324, 10.79818772836877232895590900143

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