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Properties

Label 2-1530-85.64-c1-0-19
Degree $2$
Conductor $1530$
Sign $0.980 - 0.197i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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  + 2-s + 4-s + (−2.12 + 0.707i)5-s + (1 + i)7-s + 8-s + (−2.12 + 0.707i)10-s + (4.41 − 4.41i)11-s + 3i·13-s + (1 + i)14-s + 16-s + (−2.12 − 3.53i)17-s − 1.24i·19-s + (−2.12 + 0.707i)20-s + (4.41 − 4.41i)22-s + (2.82 + 2.82i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.948 + 0.316i)5-s + (0.377 + 0.377i)7-s + 0.353·8-s + (−0.670 + 0.223i)10-s + (1.33 − 1.33i)11-s + 0.832i·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (−0.514 − 0.857i)17-s − 0.285i·19-s + (−0.474 + 0.158i)20-s + (0.941 − 0.941i)22-s + (0.589 + 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.980 - 0.197i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.980 - 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.554943632\)
\(L(\frac12)\) \(\approx\) \(2.554943632\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.12 - 0.707i)T \)
17 \( 1 + (2.12 + 3.53i)T \)
good7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 + (-4.41 + 4.41i)T - 11iT^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + 29iT^{2} \)
31 \( 1 + (-7.36 - 7.36i)T + 31iT^{2} \)
37 \( 1 + (3.24 - 3.24i)T - 37iT^{2} \)
41 \( 1 + (1.58 - 1.58i)T - 41iT^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 - 4.41iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 6.89iT - 59T^{2} \)
61 \( 1 + (-1.87 + 1.87i)T - 61iT^{2} \)
67 \( 1 + 2.48iT - 67T^{2} \)
71 \( 1 + (2.29 + 2.29i)T + 71iT^{2} \)
73 \( 1 + (-4.36 + 4.36i)T - 73iT^{2} \)
79 \( 1 + (-8.24 + 8.24i)T - 79iT^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 5.48T + 89T^{2} \)
97 \( 1 + (4.12 - 4.12i)T - 97iT^{2} \)
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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

−9.188260724875481919010401835730, −8.769210420913792182663083735536, −7.80229436500621224407194442553, −6.79762481664056852824796085798, −6.42165254505855189940161835584, −5.18906149057208488313460285435, −4.38567337351302297485285054242, −3.54108952687342968817601741672, −2.71406446014248520961863647265, −1.12682945874803436967608456919, 1.06620595708081479950075203966, 2.39936251655951801874597804376, 3.86914838001123328839438562824, 4.19619376395011741395120055859, 5.04242165848252117802515151817, 6.21555640951452683631107163319, 7.01039351600050877632651883971, 7.72207925087069180374915547157, 8.475059977771574611374940420595, 9.427685603268894074488673425246

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