Properties

Label 2-690-23.16-c1-0-5
Degree $2$
Conductor $690$
Sign $0.869 - 0.493i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (2.52 − 0.740i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (3.03 + 3.49i)11-s + (−0.654 − 0.755i)12-s + (−2.32 − 0.682i)13-s + (−1.09 + 2.39i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (0.246 − 1.71i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (−0.0580 + 0.404i)6-s + (0.953 − 0.280i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (0.914 + 1.05i)11-s + (−0.189 − 0.218i)12-s + (−0.644 − 0.189i)13-s + (−0.291 + 0.639i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (0.0597 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59261 + 0.420513i\)
\(L(\frac12)\) \(\approx\) \(1.59261 + 0.420513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-3.48 - 3.29i)T \)
good7 \( 1 + (-2.52 + 0.740i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-3.03 - 3.49i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.32 + 0.682i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.246 + 1.71i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (0.221 + 1.53i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.675 + 4.69i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-0.508 - 0.326i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (0.955 - 2.09i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-3.21 - 7.03i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.14 + 3.30i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 3.51T + 47T^{2} \)
53 \( 1 + (-7.92 + 2.32i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (3.27 + 0.962i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-1.54 - 0.991i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (0.0237 - 0.0273i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (10.3 - 11.8i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.31 + 9.16i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (4.05 + 1.18i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-3.05 + 6.68i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-6.01 + 3.86i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-0.932 - 2.04i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.29337681006221219439879679820, −9.542407461282852481825103597575, −8.827805370894520293079351496752, −7.66773879298967071117392156373, −7.30710823226515797663957397989, −6.39463453995646374866142962279, −5.11190468959820347681202030960, −4.18626342526042937555479174437, −2.55448892691533672072606011557, −1.35567438185904620379046281052, 1.25035223280734391839987606209, 2.47499075549950928948468363326, 3.74514811031517635065334821215, 4.70408480035732687344993179806, 5.79336073662879471393316871672, 7.12793596957866581547326855883, 8.184687026056960937603455516571, 8.805765608964490730081596064337, 9.320423177591768477934827207311, 10.47302107334737948609765061970

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