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Properties

Label 2-690-15.2-c1-0-0
Degree $2$
Conductor $690$
Sign $-0.943 - 0.332i$
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  + (−0.707 + 0.707i)2-s + (−0.144 − 1.72i)3-s − 1.00i·4-s + (1.25 + 1.85i)5-s + (1.32 + 1.11i)6-s + (−2.31 − 2.31i)7-s + (0.707 + 0.707i)8-s + (−2.95 + 0.499i)9-s + (−2.19 − 0.423i)10-s − 0.0826i·11-s + (−1.72 + 0.144i)12-s + (−2.92 + 2.92i)13-s + 3.27·14-s + (3.01 − 2.43i)15-s − 1.00·16-s + (−5.30 + 5.30i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.0834 − 0.996i)3-s − 0.500i·4-s + (0.560 + 0.828i)5-s + (0.539 + 0.456i)6-s + (−0.875 − 0.875i)7-s + (0.250 + 0.250i)8-s + (−0.986 + 0.166i)9-s + (−0.694 − 0.134i)10-s − 0.0249i·11-s + (−0.498 + 0.0417i)12-s + (−0.810 + 0.810i)13-s + 0.875·14-s + (0.778 − 0.627i)15-s − 0.250·16-s + (−1.28 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0179955 + 0.105159i\)
\(L(\frac12)\) \(\approx\) \(0.0179955 + 0.105159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.144 + 1.72i)T \)
5 \( 1 + (-1.25 - 1.85i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (2.31 + 2.31i)T + 7iT^{2} \)
11 \( 1 + 0.0826iT - 11T^{2} \)
13 \( 1 + (2.92 - 2.92i)T - 13iT^{2} \)
17 \( 1 + (5.30 - 5.30i)T - 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
29 \( 1 - 1.98T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + (-4.80 - 4.80i)T + 37iT^{2} \)
41 \( 1 + 0.882iT - 41T^{2} \)
43 \( 1 + (1.69 - 1.69i)T - 43iT^{2} \)
47 \( 1 + (7.63 - 7.63i)T - 47iT^{2} \)
53 \( 1 + (7.26 + 7.26i)T + 53iT^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 0.876T + 61T^{2} \)
67 \( 1 + (-0.768 - 0.768i)T + 67iT^{2} \)
71 \( 1 - 1.70iT - 71T^{2} \)
73 \( 1 + (0.265 - 0.265i)T - 73iT^{2} \)
79 \( 1 - 16.8iT - 79T^{2} \)
83 \( 1 + (8.84 + 8.84i)T + 83iT^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + (7.30 + 7.30i)T + 97iT^{2} \)
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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

−10.88152131600235096240245583070, −9.863268007359249106656884522353, −9.161637706293323892201175450720, −8.081729730619936859856898924130, −6.94571534901659666783260959353, −6.78596432510268813076303089780, −6.04374529493639084424667229547, −4.56124592217815603949696571996, −2.95038298095775691675477668208, −1.81101283844967153637370691678, 0.06130160281348716279869745246, 2.25868627599946331486060094087, 3.23693571704913633956808649914, 4.55584531568654148171572815776, 5.43469501358959507088328602010, 6.28618750546522510908077227613, 7.80404405072727829682448800556, 8.814600931421432102455890142469, 9.390178693291344358042828174450, 9.856719605375482412785139979654

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