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Properties

Label 2-1170-65.47-c1-0-23
Degree $2$
Conductor $1170$
Sign $0.194 + 0.980i$
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  − i·2-s − 4-s + (0.628 − 2.14i)5-s + 5.15·7-s + i·8-s + (−2.14 − 0.628i)10-s + (2.45 − 2.45i)11-s + (0.791 + 3.51i)13-s − 5.15i·14-s + 16-s + (1.15 − 1.15i)17-s + (−0.890 + 0.890i)19-s + (−0.628 + 2.14i)20-s + (−2.45 − 2.45i)22-s + (5.86 + 5.86i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.281 − 0.959i)5-s + 1.94·7-s + 0.353i·8-s + (−0.678 − 0.198i)10-s + (0.740 − 0.740i)11-s + (0.219 + 0.975i)13-s − 1.37i·14-s + 0.250·16-s + (0.279 − 0.279i)17-s + (−0.204 + 0.204i)19-s + (−0.140 + 0.479i)20-s + (−0.523 − 0.523i)22-s + (1.22 + 1.22i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174311031\)
\(L(\frac12)\) \(\approx\) \(2.174311031\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-0.628 + 2.14i)T \)
13 \( 1 + (-0.791 - 3.51i)T \)
good7 \( 1 - 5.15T + 7T^{2} \)
11 \( 1 + (-2.45 + 2.45i)T - 11iT^{2} \)
17 \( 1 + (-1.15 + 1.15i)T - 17iT^{2} \)
19 \( 1 + (0.890 - 0.890i)T - 19iT^{2} \)
23 \( 1 + (-5.86 - 5.86i)T + 23iT^{2} \)
29 \( 1 - 2.35iT - 29T^{2} \)
31 \( 1 + (-4.49 - 4.49i)T + 31iT^{2} \)
37 \( 1 + 6.90T + 37T^{2} \)
41 \( 1 + (4.06 + 4.06i)T + 41iT^{2} \)
43 \( 1 + (-4.25 - 4.25i)T + 43iT^{2} \)
47 \( 1 + 1.21T + 47T^{2} \)
53 \( 1 + (5.09 - 5.09i)T - 53iT^{2} \)
59 \( 1 + (7.99 + 7.99i)T + 59iT^{2} \)
61 \( 1 + 6.91T + 61T^{2} \)
67 \( 1 + 4.38iT - 67T^{2} \)
71 \( 1 + (9.29 + 9.29i)T + 71iT^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 1.67iT - 79T^{2} \)
83 \( 1 + 8.89T + 83T^{2} \)
89 \( 1 + (-2.32 - 2.32i)T + 89iT^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−9.373549442083086823451091610434, −8.877081834545190523320180661001, −8.285180108627945383797365590660, −7.33506074987590129429656998797, −5.97373722739108946072125052614, −4.97011402465238990627781558651, −4.57580856905459464601598562349, −3.38963704055077093486762811883, −1.69758567922671210155844192541, −1.29148923491646328128871867915, 1.39122608205329455304138288931, 2.66610089860499550761859359650, 4.10700979888117467239628246135, 4.89774350843969052025662961668, 5.77717282866923329868478678926, 6.74293459856170886392766221912, 7.46735657673214411476651306848, 8.195932346411948800647039047193, 8.884929033309776472982855906992, 10.05702535754967463430622323994

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