Properties

Label 2-690-23.9-c1-0-14
Degree $2$
Conductor $690$
Sign $-0.272 + 0.962i$
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  + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−0.841 − 0.540i)6-s + (1.63 − 3.57i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−6.13 − 1.80i)11-s + (−0.959 − 0.281i)12-s + (−0.723 − 1.58i)13-s + (0.559 − 3.88i)14-s + (0.654 − 0.755i)15-s + (0.415 − 0.909i)16-s + (−0.131 − 0.0847i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (−0.343 − 0.220i)6-s + (0.616 − 1.35i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (−1.85 − 0.543i)11-s + (−0.276 − 0.0813i)12-s + (−0.200 − 0.439i)13-s + (0.149 − 1.03i)14-s + (0.169 − 0.195i)15-s + (0.103 − 0.227i)16-s + (−0.0319 − 0.0205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.272 + 0.962i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09412 - 1.44729i\)
\(L(\frac12)\) \(\approx\) \(1.09412 - 1.44729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-4.11 - 2.46i)T \)
good7 \( 1 + (-1.63 + 3.57i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (6.13 + 1.80i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.723 + 1.58i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.131 + 0.0847i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-6.10 + 3.92i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (5.92 + 3.80i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.21 + 3.70i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.18 - 8.24i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.548 + 3.81i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.521 - 0.601i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 4.44T + 47T^{2} \)
53 \( 1 + (-3.34 + 7.31i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.41 - 5.29i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.01 - 1.16i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-1.51 + 0.445i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-14.1 + 4.15i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-1.67 + 1.07i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-1.30 - 2.86i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.35 - 9.45i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-11.8 - 13.6i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.522 - 3.63i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.53369646370485568253772645628, −9.739651049707045395328024792990, −7.975932371176135352157456775305, −7.56072129975911565750522891656, −6.72503268702244933040822973421, −5.41077620568921423790305873824, −4.95942347952991144558372799620, −3.52698171951569340992989088205, −2.50136717664593070981089409747, −0.804614428934913398012282893702, 2.02155636035099410916942156006, 3.15371263413645317168590476726, 4.70491035075134059414727244651, 5.26138742298578179279383838722, 5.76734330889415478677312670544, 7.21505010024521829155568279595, 8.064639873233027893446503113683, 8.977969636485101948363020557209, 9.908200471543177714922915678401, 10.88189323877663761398795055347

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