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Properties

Label 2-690-5.2-c2-0-34
Degree $2$
Conductor $690$
Sign $-0.613 + 0.789i$
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 + (2.08 − 4.54i)5-s − 2.44·6-s + (−1.45 − 1.45i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−6.62 + 2.46i)10-s + 6.56·11-s + (2.44 + 2.44i)12-s + (16.0 − 16.0i)13-s + 2.91i·14-s + (−3.01 − 8.11i)15-s − 4·16-s + (10.8 + 10.8i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.416 − 0.909i)5-s − 0.408·6-s + (−0.208 − 0.208i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.662 + 0.246i)10-s + 0.597·11-s + (0.204 + 0.204i)12-s + (1.23 − 1.23i)13-s + 0.208i·14-s + (−0.201 − 0.541i)15-s − 0.250·16-s + (0.639 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.613 + 0.789i$
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: \((2,\ 690,\ (\ :1),\ -0.613 + 0.789i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.808597528\)
\(L(\frac12)\) \(\approx\) \(1.808597528\)
\(L(2)\) 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 \( 1 + (-2.08 + 4.54i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (1.45 + 1.45i)T + 49iT^{2} \)
11 \( 1 - 6.56T + 121T^{2} \)
13 \( 1 + (-16.0 + 16.0i)T - 169iT^{2} \)
17 \( 1 + (-10.8 - 10.8i)T + 289iT^{2} \)
19 \( 1 - 8.18iT - 361T^{2} \)
29 \( 1 + 6.33iT - 841T^{2} \)
31 \( 1 - 27.7T + 961T^{2} \)
37 \( 1 + (14.1 + 14.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 69.6T + 1.68e3T^{2} \)
43 \( 1 + (-17.6 + 17.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-22.1 - 22.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (1.75 - 1.75i)T - 2.80e3iT^{2} \)
59 \( 1 - 84.6iT - 3.48e3T^{2} \)
61 \( 1 - 30.9T + 3.72e3T^{2} \)
67 \( 1 + (30.7 + 30.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 41.0T + 5.04e3T^{2} \)
73 \( 1 + (37.1 - 37.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 95.7iT - 6.24e3T^{2} \)
83 \( 1 + (-44.3 + 44.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 142. iT - 7.92e3T^{2} \)
97 \( 1 + (-99.3 - 99.3i)T + 9.40e3iT^{2} \)
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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

−10.03690500810855675693517745648, −8.919620679943184941921818982396, −8.458015707902574403130614661753, −7.67511552197452232519932649797, −6.39777786820299378252314867437, −5.55071789458825493773133734957, −4.08011318150338156249149140135, −3.17298355002318166231187587411, −1.66904039533804646922133946664, −0.800936009952168107189647594589, 1.53685415611106890549383814598, 2.91547465443564677307478543502, 3.99402726152262665989570913040, 5.31703803326378338398035583587, 6.45493264991389391924245250906, 6.87184667424617427933091748077, 8.066910769996871566089673130106, 9.014390048861963118195639984963, 9.527084453589794467436465564925, 10.37048682060920323416371664953

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