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Properties

Label 2-3510-9.4-c1-0-33
Degree $2$
Conductor $3510$
Sign $-0.766 + 0.642i$
Analytic cond. $28.0274$
Root an. cond. $5.29409$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 6·17-s + 2·19-s + (−0.499 − 0.866i)20-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + 0.999·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s + 0.316·10-s + (−0.138 + 0.240i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.45·17-s + 0.458·19-s + (−0.111 − 0.193i)20-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3510\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 13\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(28.0274\)
Root analytic conductor: \(5.29409\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3510} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3510,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5561275709\)
\(L(\frac12)\) \(\approx\) \(0.5561275709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.507956388059098215677406796066, −7.59111207774277727182869205456, −6.93263306412108027337848290074, −6.14920529501511305540640554865, −5.06246024472381815737314348866, −4.34615361725745422030423369132, −3.42989562433389133003758821818, −2.53136807947845811277760555534, −1.73501120708111865183091680574, −0.20594008064651532136076711964, 1.08864296429933281122683531749, 2.24527370673015751907489266332, 3.55603733931769555621550046886, 4.40059508050395617108056610518, 5.16677793163260823100821200612, 5.84513945826180206815584518290, 6.92708129409104874384433886132, 7.26834967997743450219227197002, 8.144755256659441386989130882212, 8.840537649475354702775836801116

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