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Properties

Label 2-690-115.68-c1-0-17
Degree $2$
Conductor $690$
Sign $-0.909 + 0.414i$
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 + (0.597 − 2.15i)5-s + 1.00·6-s + (−1.47 − 1.47i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.10 + 1.94i)10-s − 0.912i·11-s + (−0.707 + 0.707i)12-s + (−1.90 − 1.90i)13-s + 2.09·14-s + (−1.94 + 1.10i)15-s − 1.00·16-s + (−2.19 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.267 − 0.963i)5-s + 0.408·6-s + (−0.559 − 0.559i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.348 + 0.615i)10-s − 0.275i·11-s + (−0.204 + 0.204i)12-s + (−0.528 − 0.528i)13-s + 0.559·14-s + (−0.502 + 0.284i)15-s − 0.250·16-s + (−0.531 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.909 + 0.414i$
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: \((2,\ 690,\ (\ :1/2),\ -0.909 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0977395 - 0.450125i\)
\(L(\frac12)\) \(\approx\) \(0.0977395 - 0.450125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.597 + 2.15i)T \)
23 \( 1 + (1.50 - 4.55i)T \)
good7 \( 1 + (1.47 + 1.47i)T + 7iT^{2} \)
11 \( 1 + 0.912iT - 11T^{2} \)
13 \( 1 + (1.90 + 1.90i)T + 13iT^{2} \)
17 \( 1 + (2.19 + 2.19i)T + 17iT^{2} \)
19 \( 1 - 5.16T + 19T^{2} \)
29 \( 1 - 8.27iT - 29T^{2} \)
31 \( 1 + 5.45T + 31T^{2} \)
37 \( 1 + (3.73 + 3.73i)T + 37iT^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + (-0.0783 + 0.0783i)T - 43iT^{2} \)
47 \( 1 + (4.92 - 4.92i)T - 47iT^{2} \)
53 \( 1 + (-8.84 + 8.84i)T - 53iT^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 + 9.70iT - 61T^{2} \)
67 \( 1 + (5.22 + 5.22i)T + 67iT^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (8.97 + 8.97i)T + 73iT^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + (8.86 - 8.86i)T - 83iT^{2} \)
89 \( 1 + 0.837T + 89T^{2} \)
97 \( 1 + (0.302 + 0.302i)T + 97iT^{2} \)
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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

−9.881722976133462271084480037365, −9.275337351652466077090889977314, −8.320729354909431998006765737099, −7.38108240484819166378559044757, −6.74579882453512979800663511745, −5.48448071213666966012000609253, −5.06913030970276146578145271171, −3.46024006632884385344013007218, −1.62988315232822094433740108092, −0.29385634896782054193096143278, 2.06472392130697401958521295365, 3.09092262058636961290352087644, 4.19957645479452156331988182537, 5.52931018268658672901671329383, 6.51958666483348839422318041744, 7.23320495997294707399117827363, 8.465344423873370754244091462263, 9.463841549545829019171682094318, 9.995416348050599747896685036946, 10.65747394901223279969017794051

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