Properties

Label 2-690-1.1-c1-0-6
Degree $2$
Conductor $690$
Sign $1$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 3.12·7-s + 8-s + 9-s + 10-s − 3.12·11-s − 12-s + 2·13-s + 3.12·14-s − 15-s + 16-s − 1.12·17-s + 18-s + 4·19-s + 20-s − 3.12·21-s − 3.12·22-s + 23-s − 24-s + 25-s + 2·26-s − 27-s + 3.12·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.941·11-s − 0.288·12-s + 0.554·13-s + 0.834·14-s − 0.258·15-s + 0.250·16-s − 0.272·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.681·21-s − 0.665·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.590·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.331129555\)
\(L(\frac12)\) \(\approx\) \(2.331129555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 5.12T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−10.70347967111563947986137354215, −9.893093454040967770924047872289, −8.600193152708372041192707561691, −7.72790984760758369525384065297, −6.80283966174642378824177880289, −5.66120849697089272939956247000, −5.16330293180140880866797513771, −4.21035356013394717146926553200, −2.73794016074583526324836268727, −1.41641795769549472108387942963, 1.41641795769549472108387942963, 2.73794016074583526324836268727, 4.21035356013394717146926553200, 5.16330293180140880866797513771, 5.66120849697089272939956247000, 6.80283966174642378824177880289, 7.72790984760758369525384065297, 8.600193152708372041192707561691, 9.893093454040967770924047872289, 10.70347967111563947986137354215

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