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Properties

Label 2-690-115.22-c1-0-8
Degree $2$
Conductor $690$
Sign $-0.811 - 0.585i$
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 + (0.397 + 2.20i)5-s − 1.00·6-s + (1.66 − 1.66i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (−1.27 + 1.83i)10-s + 3.67i·11-s + (−0.707 − 0.707i)12-s + (−4.53 + 4.53i)13-s + 2.35·14-s + (−1.83 − 1.27i)15-s − 1.00·16-s + (4.03 − 4.03i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.177 + 0.984i)5-s − 0.408·6-s + (0.630 − 0.630i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.403 + 0.580i)10-s + 1.10i·11-s + (−0.204 − 0.204i)12-s + (−1.25 + 1.25i)13-s + 0.630·14-s + (−0.474 − 0.329i)15-s − 0.250·16-s + (0.978 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.811 - 0.585i$
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: \((2,\ 690,\ (\ :1/2),\ -0.811 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498222 + 1.54228i\)
\(L(\frac12)\) \(\approx\) \(0.498222 + 1.54228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.397 - 2.20i)T \)
23 \( 1 + (-1.01 - 4.68i)T \)
good7 \( 1 + (-1.66 + 1.66i)T - 7iT^{2} \)
11 \( 1 - 3.67iT - 11T^{2} \)
13 \( 1 + (4.53 - 4.53i)T - 13iT^{2} \)
17 \( 1 + (-4.03 + 4.03i)T - 17iT^{2} \)
19 \( 1 - 0.786T + 19T^{2} \)
29 \( 1 + 4.68iT - 29T^{2} \)
31 \( 1 + 3.96T + 31T^{2} \)
37 \( 1 + (4.59 - 4.59i)T - 37iT^{2} \)
41 \( 1 - 6.32T + 41T^{2} \)
43 \( 1 + (-1.71 - 1.71i)T + 43iT^{2} \)
47 \( 1 + (5.71 + 5.71i)T + 47iT^{2} \)
53 \( 1 + (5.93 + 5.93i)T + 53iT^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 3.15iT - 61T^{2} \)
67 \( 1 + (-3.28 + 3.28i)T - 67iT^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (-5.62 + 5.62i)T - 73iT^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-5.05 - 5.05i)T + 83iT^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (2.18 - 2.18i)T - 97iT^{2} \)
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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

−10.90087598728988714332489559319, −9.794142232000115471175405219355, −9.486186211189141504122300373140, −7.64317060246533608420095480334, −7.31970846182210560294350657588, −6.48948850753646891323688587760, −5.21843319780863899868191672805, −4.59000136231497082544348189313, −3.50372068063244503302128955980, −2.09673319209821113268431501425, 0.77172879795868340410790361746, 2.09862830658003689294387711958, 3.40912912140807420933923872127, 4.92334890844150508295427626403, 5.39876897110220724238035256220, 6.13693559327631268480886141938, 7.68697496120193153484615766678, 8.356434367926068493176382159775, 9.286203572602638081855479903868, 10.37662768965052843867120352822

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