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Properties

Label 2-1170-195.44-c1-0-5
Degree $2$
Conductor $1170$
Sign $-0.357 - 0.934i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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 − 1.00i·4-s + (1.15 − 1.91i)5-s + (−1.83 + 1.83i)7-s + (0.707 + 0.707i)8-s + (0.539 + 2.17i)10-s + (−0.354 + 0.354i)11-s + (−2.54 − 2.55i)13-s − 2.59i·14-s − 1.00·16-s + 7.26i·17-s + (−0.706 + 0.706i)19-s + (−1.91 − 1.15i)20-s − 0.500i·22-s + 6.38i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.515 − 0.856i)5-s + (−0.692 + 0.692i)7-s + (0.250 + 0.250i)8-s + (0.170 + 0.686i)10-s + (−0.106 + 0.106i)11-s + (−0.705 − 0.708i)13-s − 0.692i·14-s − 0.250·16-s + 1.76i·17-s + (−0.162 + 0.162i)19-s + (−0.428 − 0.257i)20-s − 0.106i·22-s + 1.33i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8413521410\)
\(L(\frac12)\) \(\approx\) \(0.8413521410\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.15 + 1.91i)T \)
13 \( 1 + (2.54 + 2.55i)T \)
good7 \( 1 + (1.83 - 1.83i)T - 7iT^{2} \)
11 \( 1 + (0.354 - 0.354i)T - 11iT^{2} \)
17 \( 1 - 7.26iT - 17T^{2} \)
19 \( 1 + (0.706 - 0.706i)T - 19iT^{2} \)
23 \( 1 - 6.38iT - 23T^{2} \)
29 \( 1 - 5.05iT - 29T^{2} \)
31 \( 1 + (-5.80 + 5.80i)T - 31iT^{2} \)
37 \( 1 + (6.17 - 6.17i)T - 37iT^{2} \)
41 \( 1 + (0.648 + 0.648i)T + 41iT^{2} \)
43 \( 1 - 4.80T + 43T^{2} \)
47 \( 1 + (-5.03 - 5.03i)T + 47iT^{2} \)
53 \( 1 - 1.69T + 53T^{2} \)
59 \( 1 + (-0.280 + 0.280i)T - 59iT^{2} \)
61 \( 1 - 0.831T + 61T^{2} \)
67 \( 1 + (-9.27 - 9.27i)T + 67iT^{2} \)
71 \( 1 + (-8.91 - 8.91i)T + 71iT^{2} \)
73 \( 1 + (8.01 - 8.01i)T - 73iT^{2} \)
79 \( 1 + 5.27T + 79T^{2} \)
83 \( 1 + (-1.93 + 1.93i)T - 83iT^{2} \)
89 \( 1 + (-7.14 + 7.14i)T - 89iT^{2} \)
97 \( 1 + (4.75 + 4.75i)T + 97iT^{2} \)
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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

−9.923166811273907143621638756753, −9.156656719185353929997524670958, −8.444386382683915337052618436352, −7.76892608684527849431663389020, −6.61156885142729995270331540716, −5.77966055519010049032605403296, −5.30063657319291765241361190907, −4.03385278346730848100730843869, −2.60580633334568042637898588629, −1.36272056053268679284479549933, 0.43811628893240232362208149454, 2.25848804346170170889088958715, 2.94888386208289451821967264288, 4.08322084987769562318933311382, 5.18245480005011851309547281459, 6.65423859622923220596254746643, 6.87975432130645032915471951227, 7.82153127348090029399535390664, 9.057785336336760746846579814749, 9.603549444172257210907327292790

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