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Properties

Label 2-690-115.22-c1-0-14
Degree $2$
Conductor $690$
Sign $0.630 - 0.775i$
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 + (0.643 + 2.14i)5-s + 1.00·6-s + (2.01 − 2.01i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−1.05 + 1.96i)10-s + 5.11i·11-s + (0.707 + 0.707i)12-s + (4.36 − 4.36i)13-s + 2.85·14-s + (1.96 + 1.05i)15-s − 1.00·16-s + (−0.390 + 0.390i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.287 + 0.957i)5-s + 0.408·6-s + (0.762 − 0.762i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.335 + 0.622i)10-s + 1.54i·11-s + (0.204 + 0.204i)12-s + (1.21 − 1.21i)13-s + 0.762·14-s + (0.508 + 0.273i)15-s − 0.250·16-s + (−0.0948 + 0.0948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.630 - 0.775i$
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: \((2,\ 690,\ (\ :1/2),\ 0.630 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29392 + 1.09125i\)
\(L(\frac12)\) \(\approx\) \(2.29392 + 1.09125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.643 - 2.14i)T \)
23 \( 1 + (-4.79 + 0.204i)T \)
good7 \( 1 + (-2.01 + 2.01i)T - 7iT^{2} \)
11 \( 1 - 5.11iT - 11T^{2} \)
13 \( 1 + (-4.36 + 4.36i)T - 13iT^{2} \)
17 \( 1 + (0.390 - 0.390i)T - 17iT^{2} \)
19 \( 1 + 6.15T + 19T^{2} \)
29 \( 1 - 5.15iT - 29T^{2} \)
31 \( 1 - 5.55T + 31T^{2} \)
37 \( 1 + (-1.66 + 1.66i)T - 37iT^{2} \)
41 \( 1 + 8.20T + 41T^{2} \)
43 \( 1 + (-3.51 - 3.51i)T + 43iT^{2} \)
47 \( 1 + (3.69 + 3.69i)T + 47iT^{2} \)
53 \( 1 + (3.27 + 3.27i)T + 53iT^{2} \)
59 \( 1 + 9.42iT - 59T^{2} \)
61 \( 1 + 6.90iT - 61T^{2} \)
67 \( 1 + (-6.18 + 6.18i)T - 67iT^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (9.69 - 9.69i)T - 73iT^{2} \)
79 \( 1 + 0.836T + 79T^{2} \)
83 \( 1 + (7.68 + 7.68i)T + 83iT^{2} \)
89 \( 1 + 8.61T + 89T^{2} \)
97 \( 1 + (7.13 - 7.13i)T - 97iT^{2} \)
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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

−10.69099850868529146020973087704, −9.840608888109565457217609977707, −8.493101231122701092113742043661, −7.85520492783270283276132862362, −6.94748671621378795934663328194, −6.44948054082941356255551930348, −5.13688652494939983966631437581, −4.09500323832586598197875920268, −3.02290974922352271896294330273, −1.72566711372810446895263311963, 1.34730876796241804369083936957, 2.54969320841975714620643306742, 3.91424591603524064808129819927, 4.67732705182705693276585667475, 5.68413957791610247448289800981, 6.41045238012768981752729468284, 8.369677088485868663799662334161, 8.599138641292517254756100649928, 9.301131505776357789440584779973, 10.51186346165225064511274684206

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