Properties

Label 2-690-115.22-c1-0-7
Degree $2$
Conductor $690$
Sign $-0.804 - 0.593i$
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 + (2.22 + 0.254i)5-s − 1.00·6-s + (−2.54 + 2.54i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (1.39 + 1.75i)10-s + 3.50i·11-s + (−0.707 − 0.707i)12-s + (2.40 − 2.40i)13-s − 3.60·14-s + (−1.75 + 1.39i)15-s − 1.00·16-s + (−1.37 + 1.37i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.993 + 0.113i)5-s − 0.408·6-s + (−0.962 + 0.962i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.439 + 0.553i)10-s + 1.05i·11-s + (−0.204 − 0.204i)12-s + (0.667 − 0.667i)13-s − 0.962·14-s + (−0.452 + 0.359i)15-s − 0.250·16-s + (−0.332 + 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.804 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501032 + 1.52206i\)
\(L(\frac12)\) \(\approx\) \(0.501032 + 1.52206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.254i)T \)
23 \( 1 + (4.21 - 2.27i)T \)
good7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 - 3.50iT - 11T^{2} \)
13 \( 1 + (-2.40 + 2.40i)T - 13iT^{2} \)
17 \( 1 + (1.37 - 1.37i)T - 17iT^{2} \)
19 \( 1 + 2.31T + 19T^{2} \)
29 \( 1 + 0.113iT - 29T^{2} \)
31 \( 1 - 1.27T + 31T^{2} \)
37 \( 1 + (3.74 - 3.74i)T - 37iT^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 + (-2.71 - 2.71i)T + 43iT^{2} \)
47 \( 1 + (0.775 + 0.775i)T + 47iT^{2} \)
53 \( 1 + (-2.61 - 2.61i)T + 53iT^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 + 0.841iT - 61T^{2} \)
67 \( 1 + (5.33 - 5.33i)T - 67iT^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (7.88 - 7.88i)T - 73iT^{2} \)
79 \( 1 + 2.37T + 79T^{2} \)
83 \( 1 + (1.02 + 1.02i)T + 83iT^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (-4.66 + 4.66i)T - 97iT^{2} \)
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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

−10.66139590554486046868071613072, −9.874250085380758649042680382566, −9.233421366733658293846312148326, −8.276078062617657572819084869396, −6.91492427893202439762480608661, −6.12111185876380440434097741454, −5.66618156942692839067404487151, −4.58787123698129232056540501338, −3.33556503122653040850289364200, −2.15597916881638050875357566737, 0.74234977656532535368917251590, 2.16762499451403151772989559528, 3.46915435499165423936328953912, 4.50151162427785633974223004707, 5.86364636413680743116539663046, 6.27954638940152017882685289074, 7.13897453799778319930171705674, 8.613025649002191505909499828288, 9.430344965415400058187424033596, 10.44316288255532614851631375231

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