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Properties

Label 2-1170-65.18-c1-0-15
Degree $2$
Conductor $1170$
Sign $0.121 - 0.992i$
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 + (2.18 + 0.482i)5-s + 2.10·7-s − i·8-s + (−0.482 + 2.18i)10-s + (1.24 + 1.24i)11-s + (−3.37 + 1.25i)13-s + 2.10i·14-s + 16-s + (1.54 + 1.54i)17-s + (1.19 + 1.19i)19-s + (−2.18 − 0.482i)20-s + (−1.24 + 1.24i)22-s + (2.88 − 2.88i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.976 + 0.215i)5-s + 0.796·7-s − 0.353i·8-s + (−0.152 + 0.690i)10-s + (0.374 + 0.374i)11-s + (−0.937 + 0.348i)13-s + 0.563i·14-s + 0.250·16-s + (0.373 + 0.373i)17-s + (0.274 + 0.274i)19-s + (−0.488 − 0.107i)20-s + (−0.264 + 0.264i)22-s + (0.600 − 0.600i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.121 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.010865913\)
\(L(\frac12)\) \(\approx\) \(2.010865913\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.482i)T \)
13 \( 1 + (3.37 - 1.25i)T \)
good7 \( 1 - 2.10T + 7T^{2} \)
11 \( 1 + (-1.24 - 1.24i)T + 11iT^{2} \)
17 \( 1 + (-1.54 - 1.54i)T + 17iT^{2} \)
19 \( 1 + (-1.19 - 1.19i)T + 19iT^{2} \)
23 \( 1 + (-2.88 + 2.88i)T - 23iT^{2} \)
29 \( 1 + 0.655iT - 29T^{2} \)
31 \( 1 + (5.01 - 5.01i)T - 31iT^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 + (-5.07 + 5.07i)T - 41iT^{2} \)
43 \( 1 + (3.01 - 3.01i)T - 43iT^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 + (3.88 + 3.88i)T + 53iT^{2} \)
59 \( 1 + (-1.87 + 1.87i)T - 59iT^{2} \)
61 \( 1 - 4.35T + 61T^{2} \)
67 \( 1 - 7.11iT - 67T^{2} \)
71 \( 1 + (1.46 - 1.46i)T - 71iT^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 1.75iT - 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + (-1.31 + 1.31i)T - 89iT^{2} \)
97 \( 1 - 6.58iT - 97T^{2} \)
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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

−9.798569783932762373508427192832, −9.159612909866021817238178548273, −8.301790188515535310357223299883, −7.33234962486840577905531377280, −6.74321395936533106972560622803, −5.70188941170332486361384762300, −5.05626046148250977940370422885, −4.11311462902459962697585219277, −2.61114373192204733821753376314, −1.43241936094126426826100236921, 1.00319686212874187154002620173, 2.13244754830504347311662466894, 3.10031621602421912698115146376, 4.45610139480457382912243347146, 5.23293742845569466229620613908, 5.94999255675607729033547740990, 7.25143097842195586865904374058, 8.039378343379673835411745922545, 9.164879260504983584293797653771, 9.490252749849414923521223195978

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