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Properties

Label 2-1169-1169.333-c0-0-4
Degree $2$
Conductor $1169$
Sign $0.954 + 0.296i$
Analytic cond. $0.583406$
Root an. cond. $0.763810$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 − 1.45i)2-s + (−0.981 + 1.70i)3-s + (−0.915 + 1.58i)4-s + 3.30·6-s + (0.723 + 0.690i)7-s + 1.39·8-s + (−1.42 − 2.47i)9-s + (0.995 − 1.72i)11-s + (−1.79 − 3.11i)12-s + (0.396 − 1.63i)14-s + (−0.260 − 0.451i)16-s + (−2.40 + 4.16i)18-s + (0.142 + 0.246i)19-s + (−1.88 + 0.553i)21-s − 3.34·22-s + ⋯
L(s)  = 1  + (−0.841 − 1.45i)2-s + (−0.981 + 1.70i)3-s + (−0.915 + 1.58i)4-s + 3.30·6-s + (0.723 + 0.690i)7-s + 1.39·8-s + (−1.42 − 2.47i)9-s + (0.995 − 1.72i)11-s + (−1.79 − 3.11i)12-s + (0.396 − 1.63i)14-s + (−0.260 − 0.451i)16-s + (−2.40 + 4.16i)18-s + (0.142 + 0.246i)19-s + (−1.88 + 0.553i)21-s − 3.34·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(0.583406\)
Root analytic conductor: \(0.763810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1169,\ (\ :0),\ 0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5054924669\)
\(L(\frac12)\) \(\approx\) \(0.5054924669\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.723 - 0.690i)T \)
167 \( 1 - T \)
good2 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 0.471T + T^{2} \)
31 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.654 - 1.13i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−10.02796974621166590146786133213, −9.316125843716177481059209685277, −8.868042783126008359948340296937, −8.125932562808299052995250765903, −6.14263943117228735790494741506, −5.63132975487135040750852401437, −4.40309114064672471161787530650, −3.70747700011567340443688534520, −2.82525829483672272533036080965, −0.992770933656662698523989163583, 0.993782236146804785311703723286, 1.98634767357860390441959270933, 4.59473336071029419848803116290, 5.25298518332197740296154989250, 6.39224404104692308153532767386, 6.83483531300856952651212186433, 7.33128214133968410115997088649, 8.022343831943800423296974618671, 8.725690250511854968661160183828, 9.961460626098602963047514395841

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