Properties

Label 2-690-23.9-c1-0-15
Degree $2$
Conductor $690$
Sign $-0.298 + 0.954i$
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 + (1.24 − 2.72i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (1.52 + 0.449i)11-s + (−0.959 − 0.281i)12-s + (−0.404 − 0.886i)13-s + (0.425 − 2.96i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (−0.675 − 0.434i)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 + (0.469 − 1.02i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.131 − 0.287i)10-s + (0.461 + 0.135i)11-s + (−0.276 − 0.0813i)12-s + (−0.112 − 0.245i)13-s + (0.113 − 0.791i)14-s + (−0.169 + 0.195i)15-s + (0.103 − 0.227i)16-s + (−0.163 − 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.298 + 0.954i$
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: \((2,\ 690,\ (\ :1/2),\ -0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15707 - 1.57404i\)
\(L(\frac12)\) \(\approx\) \(1.15707 - 1.57404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.04 + 2.56i)T \)
good7 \( 1 + (-1.24 + 2.72i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-1.52 - 0.449i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.404 + 0.886i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.675 + 0.434i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (1.37 - 0.883i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (1.50 + 0.966i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.90 + 4.51i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.999 + 6.95i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.899 - 6.25i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (1.02 + 1.17i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 + (0.819 - 1.79i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.74 - 6.00i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.52 - 1.75i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-10.7 + 3.14i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-4.80 + 1.41i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.34 - 2.15i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-0.632 - 1.38i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.99 - 13.8i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-3.91 - 4.51i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.0304 - 0.211i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.46445332223939340753668078230, −9.538407840885782818765067283123, −8.233072156914722567310263194267, −7.49865985885092074693310552852, −6.56175273674023951876701247843, −5.66391504320418170705155433265, −4.54681322415723710698139122875, −3.92283675624449351832336252669, −2.23930663725777000872510623437, −0.901143949634322593195259186338, 2.03237651275780872326966992607, 3.29739476760042066722459737132, 4.38183674244321234823082716636, 5.31529844912737128125558709392, 6.13051765395085676596305277941, 6.93512059187186156774290626754, 8.137331932469262428943747455038, 8.955523484231972492638834812896, 9.979570112577498750533978798554, 10.92126627068810753536220886907

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