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Properties

Label 2-690-5.3-c2-0-31
Degree $2$
Conductor $690$
Sign $-0.315 + 0.949i$
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 + (2.62 − 4.25i)5-s − 2.44·6-s + (0.185 − 0.185i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−1.63 − 6.87i)10-s + 17.6·11-s + (−2.44 + 2.44i)12-s + (16.7 + 16.7i)13-s − 0.371i·14-s + (−8.42 + 1.99i)15-s − 4·16-s + (19.8 − 19.8i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.524 − 0.851i)5-s − 0.408·6-s + (0.0265 − 0.0265i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.163 − 0.687i)10-s + 1.60·11-s + (−0.204 + 0.204i)12-s + (1.29 + 1.29i)13-s − 0.0265i·14-s + (−0.561 + 0.133i)15-s − 0.250·16-s + (1.16 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.315 + 0.949i$
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: \((2,\ 690,\ (\ :1),\ -0.315 + 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.674738962\)
\(L(\frac12)\) \(\approx\) \(2.674738962\)
\(L(2)\) 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 \( 1 + (-2.62 + 4.25i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (-0.185 + 0.185i)T - 49iT^{2} \)
11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + (-16.7 - 16.7i)T + 169iT^{2} \)
17 \( 1 + (-19.8 + 19.8i)T - 289iT^{2} \)
19 \( 1 + 16.2iT - 361T^{2} \)
29 \( 1 - 5.94iT - 841T^{2} \)
31 \( 1 + 4.19T + 961T^{2} \)
37 \( 1 + (45.9 - 45.9i)T - 1.36e3iT^{2} \)
41 \( 1 + 37.8T + 1.68e3T^{2} \)
43 \( 1 + (24.5 + 24.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-30.1 + 30.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (54.4 + 54.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 87.7iT - 3.48e3T^{2} \)
61 \( 1 - 20.2T + 3.72e3T^{2} \)
67 \( 1 + (-5.03 + 5.03i)T - 4.48e3iT^{2} \)
71 \( 1 - 42.6T + 5.04e3T^{2} \)
73 \( 1 + (-25.5 - 25.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 85.8iT - 6.24e3T^{2} \)
83 \( 1 + (49.0 + 49.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 64.8iT - 7.92e3T^{2} \)
97 \( 1 + (63.8 - 63.8i)T - 9.40e3iT^{2} \)
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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

−9.980416353776414095759408407037, −9.171650490069649455954742333249, −8.607645725386417965153877220196, −6.99583634453870335230321748285, −6.38653016629382179365428612474, −5.37644604516027063652544346465, −4.51496306779541678037729946896, −3.44365606207540222939954515833, −1.72850327150084562643628628987, −1.02544223783884354737759588062, 1.47925893492772700381752704262, 3.45108149548858308379431254638, 3.76087552666162247752554062496, 5.41336530373702234918046468000, 6.04891916042815212340850868988, 6.62269607719035177978657720789, 7.83367837402688569487565202883, 8.724924895955415175063317989027, 9.821138039908671422345576327482, 10.55648011456734479957241476449

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