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Properties

Label 2-1170-195.44-c1-0-25
Degree $2$
Conductor $1170$
Sign $-0.847 + 0.530i$
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 + (1.91 − 1.15i)5-s + (−1.29 + 1.29i)7-s + (0.707 + 0.707i)8-s + (−0.539 + 2.17i)10-s + (−3.42 + 3.42i)11-s + (−3.25 − 1.55i)13-s − 1.82i·14-s − 1.00·16-s − 4.71i·17-s + (−3.83 + 3.83i)19-s + (−1.15 − 1.91i)20-s − 4.84i·22-s − 3.20i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.856 − 0.515i)5-s + (−0.488 + 0.488i)7-s + (0.250 + 0.250i)8-s + (−0.170 + 0.686i)10-s + (−1.03 + 1.03i)11-s + (−0.902 − 0.431i)13-s − 0.488i·14-s − 0.250·16-s − 1.14i·17-s + (−0.879 + 0.879i)19-s + (−0.257 − 0.428i)20-s − 1.03i·22-s − 0.668i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.847 + 0.530i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07293317225\)
\(L(\frac12)\) \(\approx\) \(0.07293317225\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 1.15i)T \)
13 \( 1 + (3.25 + 1.55i)T \)
good7 \( 1 + (1.29 - 1.29i)T - 7iT^{2} \)
11 \( 1 + (3.42 - 3.42i)T - 11iT^{2} \)
17 \( 1 + 4.71iT - 17T^{2} \)
19 \( 1 + (3.83 - 3.83i)T - 19iT^{2} \)
23 \( 1 + 3.20iT - 23T^{2} \)
29 \( 1 + 7.09iT - 29T^{2} \)
31 \( 1 + (4.09 - 4.09i)T - 31iT^{2} \)
37 \( 1 + (3.72 - 3.72i)T - 37iT^{2} \)
41 \( 1 + (-4.42 - 4.42i)T + 41iT^{2} \)
43 \( 1 + 7.68T + 43T^{2} \)
47 \( 1 + (8.97 + 8.97i)T + 47iT^{2} \)
53 \( 1 - 6.86T + 53T^{2} \)
59 \( 1 + (1.67 - 1.67i)T - 59iT^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (5.61 + 5.61i)T + 67iT^{2} \)
71 \( 1 + (-3.05 - 3.05i)T + 71iT^{2} \)
73 \( 1 + (0.453 - 0.453i)T - 73iT^{2} \)
79 \( 1 - 8.29T + 79T^{2} \)
83 \( 1 + (2.48 - 2.48i)T - 83iT^{2} \)
89 \( 1 + (3.36 - 3.36i)T - 89iT^{2} \)
97 \( 1 + (3.71 + 3.71i)T + 97iT^{2} \)
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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

−9.610886362745310069094137257624, −8.624190932922583019979997650065, −7.86997005391504658843584785580, −6.96560602458570618745156751031, −6.11315455307801772641254172526, −5.21928129846523639931340558677, −4.65810704622017128980086791771, −2.77115898098556408488209924620, −1.93060201320649935368267876063, −0.03351759359098162403683427510, 1.79102668270345583130925261038, 2.79978495064715708317848102637, 3.67740503015836432737075014660, 5.03468606062117233192861456945, 6.01938082690150303079514169273, 6.88649643681773533085300932361, 7.65762971486966398736135473685, 8.713129764505218601158813702224, 9.373327226256448334208615045798, 10.26369491286949211734423475252

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