Properties

Label 2-690-23.16-c1-0-11
Degree $2$
Conductor $690$
Sign $0.0243 + 0.999i$
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 + (4.34 − 1.27i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (2.77 + 3.19i)11-s + (−0.654 − 0.755i)12-s + (−2.94 − 0.863i)13-s + (1.88 − 4.11i)14-s + (−0.841 − 0.540i)15-s + (−0.959 + 0.281i)16-s + (0.681 − 4.74i)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 + (1.64 − 0.482i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (0.835 + 0.963i)11-s + (−0.189 − 0.218i)12-s + (−0.815 − 0.239i)13-s + (0.502 − 1.10i)14-s + (−0.217 − 0.139i)15-s + (−0.239 + 0.0704i)16-s + (0.165 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0243 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0243 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0243 + 0.999i$
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: \((2,\ 690,\ (\ :1/2),\ 0.0243 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79772 - 1.75443i\)
\(L(\frac12)\) \(\approx\) \(1.79772 - 1.75443i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.54 - 1.51i)T \)
good7 \( 1 + (-4.34 + 1.27i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-2.77 - 3.19i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.94 + 0.863i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.681 + 4.74i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.841 - 5.85i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.738 + 5.13i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-2.06 - 1.32i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.64 - 3.60i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (2.62 + 5.74i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-0.112 + 0.0721i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + (3.70 - 1.08i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-9.79 - 2.87i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-4.18 - 2.68i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-5.43 + 6.27i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (7.92 - 9.14i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.910 - 6.33i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-10.5 - 3.10i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (2.17 - 4.76i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (11.5 - 7.44i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.99 + 6.54i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.10617949886774272661978506414, −9.662880939253014261921063865574, −8.366160092888068786244384860498, −7.75718557577212754186957002990, −6.88902416016210931841468016771, −5.38314858095161765229132953842, −4.59955693505692355085397649090, −3.80372264235548978567794703114, −2.20345681785410676771341780832, −1.30276551627062836582087338764, 1.92405305505588916356954032506, 3.23436744196199601808698158939, 4.35659282025759010150031175822, 5.10033202251905168890720074888, 6.20883518681890803420628060617, 7.21252777385554930918621540678, 8.281584052902044181227058533931, 8.531671628710682838299442516133, 9.677882962320216211644484782633, 10.92324958477442294206024678666

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