Properties

Label 2-690-69.68-c1-0-24
Degree $2$
Conductor $690$
Sign $-0.773 + 0.633i$
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  i·2-s + (−0.0386 + 1.73i)3-s − 4-s − 5-s + (1.73 + 0.0386i)6-s − 2.83i·7-s + i·8-s + (−2.99 − 0.133i)9-s + i·10-s + 4.50·11-s + (0.0386 − 1.73i)12-s − 5.89·13-s − 2.83·14-s + (0.0386 − 1.73i)15-s + 16-s − 5.40·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.0223 + 0.999i)3-s − 0.5·4-s − 0.447·5-s + (0.706 + 0.0157i)6-s − 1.07i·7-s + 0.353i·8-s + (−0.999 − 0.0446i)9-s + 0.316i·10-s + 1.35·11-s + (0.0111 − 0.499i)12-s − 1.63·13-s − 0.758·14-s + (0.00998 − 0.447i)15-s + 0.250·16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.773 + 0.633i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204512 - 0.572305i\)
\(L(\frac12)\) \(\approx\) \(0.204512 - 0.572305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.0386 - 1.73i)T \)
5 \( 1 + T \)
23 \( 1 + (-3.64 + 3.12i)T \)
good7 \( 1 + 2.83iT - 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + 5.89T + 13T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
29 \( 1 + 4.81iT - 29T^{2} \)
31 \( 1 + 9.92T + 31T^{2} \)
37 \( 1 + 3.92iT - 37T^{2} \)
41 \( 1 + 2.24iT - 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 - 1.55iT - 47T^{2} \)
53 \( 1 + 6.08T + 53T^{2} \)
59 \( 1 - 10.8iT - 59T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 - 3.47iT - 67T^{2} \)
71 \( 1 - 9.84iT - 71T^{2} \)
73 \( 1 - 0.323T + 73T^{2} \)
79 \( 1 + 5.69iT - 79T^{2} \)
83 \( 1 + 4.31T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 8.21iT - 97T^{2} \)
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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

−10.28742418252012923648233862600, −9.267162251348528641767590702823, −8.929144490936282735078948056569, −7.50952109566920353523399323570, −6.69226360754621308796093225223, −5.13176787688223624580938052757, −4.28299621319126286966355278627, −3.76829503962840685755267969658, −2.38454817147428606181041048828, −0.31263137503998909847262751653, 1.77033546211367368329149103878, 3.13688732299451596212772920228, 4.64516720178789043336166194223, 5.59910629179843406290293742666, 6.60550584981390754842384419991, 7.16127683559827226700276519705, 8.072913234472600017159165297636, 9.027164673841798530271130359160, 9.414439279196930096518616728234, 11.10404100714404599028413670421

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