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Properties

Label 2-1169-1169.333-c0-0-6
Degree $2$
Conductor $1169$
Sign $0.963 - 0.266i$
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  + (0.142 + 0.246i)2-s + (−0.0475 + 0.0824i)3-s + (0.459 − 0.795i)4-s − 0.0270·6-s + (0.981 + 0.189i)7-s + 0.546·8-s + (0.495 + 0.858i)9-s + (−0.723 + 1.25i)11-s + (0.0437 + 0.0757i)12-s + (0.0930 + 0.268i)14-s + (−0.381 − 0.661i)16-s + (−0.141 + 0.244i)18-s + (−0.415 − 0.719i)19-s + (−0.0623 + 0.0719i)21-s − 0.411·22-s + ⋯
L(s)  = 1  + (0.142 + 0.246i)2-s + (−0.0475 + 0.0824i)3-s + (0.459 − 0.795i)4-s − 0.0270·6-s + (0.981 + 0.189i)7-s + 0.546·8-s + (0.495 + 0.858i)9-s + (−0.723 + 1.25i)11-s + (0.0437 + 0.0757i)12-s + (0.0930 + 0.268i)14-s + (−0.381 − 0.661i)16-s + (−0.141 + 0.244i)18-s + (−0.415 − 0.719i)19-s + (−0.0623 + 0.0719i)21-s − 0.411·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.963 - 0.266i$
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: \((2,\ 1169,\ (\ :0),\ 0.963 - 0.266i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.362309665\)
\(L(\frac12)\) \(\approx\) \(1.362309665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.981 - 0.189i)T \)
167 \( 1 - T \)
good2 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 0.654T + T^{2} \)
31 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.91T + T^{2} \)
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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

−10.02958854177800319600795435890, −9.433981475186747204938165036910, −8.079439060764896151424881878384, −7.53876837630247187283991690673, −6.78822318533864624484517890428, −5.58442855692027145419300372911, −4.96428907754995373412756487135, −4.32242898379159122112768623137, −2.36205133458472624466118655003, −1.74413061951590468253205116935, 1.46577241375150951201930867310, 2.78108389424598860294457024676, 3.73265377983259358240225857296, 4.57423609925580943933369765341, 5.79604154169191281293790463450, 6.64668485610094655536631354563, 7.64754093498933639584469416999, 8.197694332573074384173926382791, 8.907796752937072700628863107981, 10.23698590028759899726433861344

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