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Properties

Label 2-3510-9.7-c1-0-5
Degree $2$
Conductor $3510$
Sign $0.173 - 0.984i$
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 + (−1 + 1.73i)7-s − 0.999·8-s − 0.999·10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s − 5·17-s + 3·19-s + (−0.499 + 0.866i)20-s + (−0.499 − 0.866i)22-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 + (−0.377 + 0.654i)7-s − 0.353·8-s − 0.316·10-s + (0.150 − 0.261i)11-s + (0.138 + 0.240i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s − 1.21·17-s + 0.688·19-s + (−0.111 + 0.193i)20-s + (−0.106 − 0.184i)22-s + (−0.0999 + 0.173i)25-s + 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3510\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 13\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(28.0274\)
Root analytic conductor: \(5.29409\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3510} (2341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3510,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6605326155\)
\(L(\frac12)\) \(\approx\) \(0.6605326155\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-3.5 - 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.836378011511291963881294864645, −8.212886836932968270747120215996, −7.22345899988281511316687118548, −6.24563690617456204549495509686, −5.77545485277356540238842344107, −4.71347161161108946780578017702, −4.17996937250630235286214880096, −3.12908415246417627941448169128, −2.38271564793668515133924981672, −1.23953861718959448476971005089, 0.17790834389363177406800796806, 1.81689252393259546455818039967, 3.18883287848661479335573067989, 3.66493443569365931540491263894, 4.68573705182230860107779556324, 5.31925618873662863103241807560, 6.38208229646195204086821196752, 7.01677364232228273208716711719, 7.29595125668086425625171373231, 8.431420282665371087248460688158

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