Properties

Label 2-690-115.33-c1-0-7
Degree $2$
Conductor $690$
Sign $0.953 - 0.300i$
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  + (−0.997 − 0.0713i)2-s + (−0.212 − 0.977i)3-s + (0.989 + 0.142i)4-s + (1.62 − 1.53i)5-s + (0.142 + 0.989i)6-s + (1.83 + 3.35i)7-s + (−0.977 − 0.212i)8-s + (−0.909 + 0.415i)9-s + (−1.73 + 1.41i)10-s + (1.05 + 0.913i)11-s + (−0.0713 − 0.997i)12-s + (4.41 + 2.41i)13-s + (−1.58 − 3.47i)14-s + (−1.84 − 1.26i)15-s + (0.959 + 0.281i)16-s + (−4.62 + 6.18i)17-s + ⋯
L(s)  = 1  + (−0.705 − 0.0504i)2-s + (−0.122 − 0.564i)3-s + (0.494 + 0.0711i)4-s + (0.726 − 0.686i)5-s + (0.0580 + 0.404i)6-s + (0.692 + 1.26i)7-s + (−0.345 − 0.0751i)8-s + (−0.303 + 0.138i)9-s + (−0.547 + 0.447i)10-s + (0.317 + 0.275i)11-s + (−0.0205 − 0.287i)12-s + (1.22 + 0.668i)13-s + (−0.424 − 0.929i)14-s + (−0.476 − 0.325i)15-s + (0.239 + 0.0704i)16-s + (−1.12 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.953 - 0.300i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25452 + 0.192946i\)
\(L(\frac12)\) \(\approx\) \(1.25452 + 0.192946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.997 + 0.0713i)T \)
3 \( 1 + (0.212 + 0.977i)T \)
5 \( 1 + (-1.62 + 1.53i)T \)
23 \( 1 + (-4.78 + 0.367i)T \)
good7 \( 1 + (-1.83 - 3.35i)T + (-3.78 + 5.88i)T^{2} \)
11 \( 1 + (-1.05 - 0.913i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-4.41 - 2.41i)T + (7.02 + 10.9i)T^{2} \)
17 \( 1 + (4.62 - 6.18i)T + (-4.78 - 16.3i)T^{2} \)
19 \( 1 + (0.753 - 5.23i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (5.88 - 0.846i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (8.20 - 5.27i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-0.492 + 1.31i)T + (-27.9 - 24.2i)T^{2} \)
41 \( 1 + (-2.71 + 5.94i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-4.16 + 0.905i)T + (39.1 - 17.8i)T^{2} \)
47 \( 1 + (-7.10 + 7.10i)T - 47iT^{2} \)
53 \( 1 + (-2.49 + 1.36i)T + (28.6 - 44.5i)T^{2} \)
59 \( 1 + (1.11 + 3.78i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (1.03 + 1.60i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (0.824 - 11.5i)T + (-66.3 - 9.53i)T^{2} \)
71 \( 1 + (2.57 + 2.96i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-6.27 + 4.69i)T + (20.5 - 70.0i)T^{2} \)
79 \( 1 + (-7.27 + 2.13i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-8.24 - 3.07i)T + (62.7 + 54.3i)T^{2} \)
89 \( 1 + (0.749 + 0.481i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (4.59 - 1.71i)T + (73.3 - 63.5i)T^{2} \)
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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

−10.67201103400112314076507322915, −9.231635442942625862649445499077, −8.797566740888039853465663123279, −8.309857228216964431910947741010, −6.98681867093387112566589535411, −6.00081733558731831390665834035, −5.48198050939191037787985831032, −3.95269273050408049347657386243, −1.99445749027928597663110543422, −1.64384597934323610388390166566, 0.927410715352264593738972344606, 2.57948994342403338050568595192, 3.80471651239714524641493007416, 4.99988272366741719766450579544, 6.11647872077504914537643773978, 7.04007005823928014172194997884, 7.71267097705688706854042942355, 9.141008355893616543484380208810, 9.335221370111490564636245338404, 10.73648781192558874595441375726

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