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Properties

Label 2-690-15.8-c1-0-42
Degree $2$
Conductor $690$
Sign $-0.374 + 0.927i$
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 + (−0.292 − 1.70i)3-s + 1.00i·4-s − 2.23i·5-s + (0.999 − 1.41i)6-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s + (1.58 − 1.58i)10-s − 2.82i·11-s + (1.70 − 0.292i)12-s + (−2.16 − 2.16i)13-s + (−3.81 + 0.654i)15-s − 1.00·16-s + (−0.821 − 0.821i)17-s + (−2.70 − 1.29i)18-s − 1.16i·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.169 − 0.985i)3-s + 0.500i·4-s − 0.999i·5-s + (0.408 − 0.577i)6-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s + (0.500 − 0.500i)10-s − 0.852i·11-s + (0.492 − 0.0845i)12-s + (−0.599 − 0.599i)13-s + (−0.985 + 0.169i)15-s − 0.250·16-s + (−0.199 − 0.199i)17-s + (−0.638 − 0.304i)18-s − 0.266i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755504 - 1.11969i\)
\(L(\frac12)\) \(\approx\) \(0.755504 - 1.11969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (2.16 + 2.16i)T + 13iT^{2} \)
17 \( 1 + (0.821 + 0.821i)T + 17iT^{2} \)
19 \( 1 + 1.16iT - 19T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + (1.16 - 1.16i)T - 37iT^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + (9.16 + 9.16i)T + 43iT^{2} \)
47 \( 1 + (-0.229 - 0.229i)T + 47iT^{2} \)
53 \( 1 + (-6.70 + 6.70i)T - 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (1.16 - 1.16i)T - 67iT^{2} \)
71 \( 1 - 5.88iT - 71T^{2} \)
73 \( 1 + (-5.32 - 5.32i)T + 73iT^{2} \)
79 \( 1 + 3.16iT - 79T^{2} \)
83 \( 1 + (-3.05 + 3.05i)T - 83iT^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 + (5.16 - 5.16i)T - 97iT^{2} \)
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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

−10.24114277739853366653021703604, −8.874534899717911105575796065705, −8.402987683565880276463728732576, −7.49849841101023184679755106028, −6.65399288370889914282650264087, −5.56203483601799481019394858842, −5.11025995794410765913879725496, −3.65766338503335376828317630412, −2.28486817663019095808799361808, −0.59463492846513305198195169951, 2.18574697193643455854648703239, 3.23993514281653328672569629859, 4.24095318563335927402088242370, 5.02355818160090036141290370588, 6.20451622000148881908182145553, 6.96307375636678353371875866086, 8.242862215789945272657056773459, 9.520911190347191011917387572701, 9.987796652696502732147414851090, 10.68513728144837370543400063251

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