Properties

Label 2-690-23.6-c1-0-1
Degree $2$
Conductor $690$
Sign $0.924 - 0.381i$
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  + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.654 + 0.755i)6-s + (−0.498 + 3.46i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (1.29 + 2.84i)11-s + (0.415 + 0.909i)12-s + (−0.133 − 0.930i)13-s + (2.94 + 1.89i)14-s + (−0.959 + 0.281i)15-s + (−0.142 + 0.989i)16-s + (−4.81 + 5.56i)17-s + ⋯
L(s)  = 1  + (0.293 − 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.376 − 0.241i)5-s + (−0.267 + 0.308i)6-s + (−0.188 + 1.30i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (0.391 + 0.857i)11-s + (0.119 + 0.262i)12-s + (−0.0371 − 0.258i)13-s + (0.786 + 0.505i)14-s + (−0.247 + 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (−1.16 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.924 - 0.381i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26897 + 0.251675i\)
\(L(\frac12)\) \(\approx\) \(1.26897 + 0.251675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-3.42 - 3.35i)T \)
good7 \( 1 + (0.498 - 3.46i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-1.29 - 2.84i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.133 + 0.930i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (4.81 - 5.56i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-1.23 - 1.42i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (0.455 - 0.525i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-4.98 + 1.46i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (0.958 + 0.615i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-1.73 + 1.11i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-9.55 - 2.80i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 + (0.573 - 3.98i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (1.82 + 12.7i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (2.91 - 0.854i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-1.53 + 3.35i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (1.92 - 4.20i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (0.931 + 1.07i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-2.40 - 16.7i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-5.18 - 3.32i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (3.33 + 0.979i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (4.25 - 2.73i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.68065827658672881186635220517, −9.643938552967243114454012038389, −9.103796580841197014041435900369, −8.076551431613354296142375863445, −6.66399392287795518239295775293, −5.91179871621247842179080703452, −5.10724278536392837199520678524, −4.08012049480695976077926445896, −2.58007897373578894046254865556, −1.58248988108016105213600443754, 0.69964990870935352868308967651, 2.92448314154834532014461455438, 4.17659404539778015041688984783, 4.90851133208064224188845117761, 6.11222870811911575389660965301, 6.82552554143344335678350144444, 7.39982278301844518134991635429, 8.753392502084128537743546734581, 9.493826555244796795561971661062, 10.53733087552777706490352433987

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