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Properties

Label 2-690-115.68-c1-0-6
Degree $2$
Conductor $690$
Sign $-0.510 - 0.860i$
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 + (0.707 + 0.707i)3-s − 1.00i·4-s + (−0.299 + 2.21i)5-s − 1.00·6-s + (3.52 + 3.52i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−1.35 − 1.77i)10-s − 3.55i·11-s + (0.707 − 0.707i)12-s + (1.85 + 1.85i)13-s − 4.98·14-s + (−1.77 + 1.35i)15-s − 1.00·16-s + (2.03 + 2.03i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.133 + 0.990i)5-s − 0.408·6-s + (1.33 + 1.33i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.428 − 0.562i)10-s − 1.07i·11-s + (0.204 − 0.204i)12-s + (0.513 + 0.513i)13-s − 1.33·14-s + (−0.459 + 0.349i)15-s − 0.250·16-s + (0.493 + 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.510 - 0.860i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.510 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748167 + 1.31355i\)
\(L(\frac12)\) \(\approx\) \(0.748167 + 1.31355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.299 - 2.21i)T \)
23 \( 1 + (4.50 + 1.63i)T \)
good7 \( 1 + (-3.52 - 3.52i)T + 7iT^{2} \)
11 \( 1 + 3.55iT - 11T^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 + (-2.03 - 2.03i)T + 17iT^{2} \)
19 \( 1 - 4.34T + 19T^{2} \)
29 \( 1 + 9.31iT - 29T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 + (-5.97 - 5.97i)T + 37iT^{2} \)
41 \( 1 + 0.173T + 41T^{2} \)
43 \( 1 + (1.69 - 1.69i)T - 43iT^{2} \)
47 \( 1 + (-4.77 + 4.77i)T - 47iT^{2} \)
53 \( 1 + (4.53 - 4.53i)T - 53iT^{2} \)
59 \( 1 - 8.64iT - 59T^{2} \)
61 \( 1 + 5.58iT - 61T^{2} \)
67 \( 1 + (6.08 + 6.08i)T + 67iT^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + (7.81 + 7.81i)T + 73iT^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + (4.22 - 4.22i)T - 83iT^{2} \)
89 \( 1 + 6.69T + 89T^{2} \)
97 \( 1 + (9.17 + 9.17i)T + 97iT^{2} \)
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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

−10.74075990821160125615736261276, −9.758582262518946326537485397150, −8.880883964238950904935557606371, −8.138175752327820389214946343545, −7.64445554452694519994065317561, −6.11311994068380677800865869598, −5.70060181826994937222452381723, −4.34077532032883459934539416060, −3.02801977228589622967222725888, −1.85504743040337848904946662812, 1.01637680272223540567164095514, 1.78098104103346944340052453877, 3.56555274673840178401691355751, 4.48282452935603220718486840003, 5.42384588611069851513504593715, 7.35998386088357208769633868237, 7.53094172309876506715922312302, 8.400644307253809191902331144327, 9.313552661824081951292596648911, 10.12750629558928135265852342088

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