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Properties

Label 2-1191-1191.401-c0-0-0
Degree $2$
Conductor $1191$
Sign $-0.417 - 0.908i$
Analytic cond. $0.594386$
Root an. cond. $0.770964$
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  + (0.415 + 0.909i)3-s + (0.142 + 0.989i)4-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)12-s + (−1.07 + 0.153i)13-s + (−0.959 + 0.281i)16-s + (1.25 + 0.368i)19-s + (0.142 + 0.989i)25-s + (−0.959 − 0.281i)27-s + (0.698 − 1.53i)31-s + (−0.841 − 0.540i)36-s + (0.797 − 1.74i)37-s + (−0.584 − 0.909i)39-s + (−0.857 + 0.989i)43-s + (−0.654 − 0.755i)48-s + (−0.142 + 0.989i)49-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (0.142 + 0.989i)4-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)12-s + (−1.07 + 0.153i)13-s + (−0.959 + 0.281i)16-s + (1.25 + 0.368i)19-s + (0.142 + 0.989i)25-s + (−0.959 − 0.281i)27-s + (0.698 − 1.53i)31-s + (−0.841 − 0.540i)36-s + (0.797 − 1.74i)37-s + (−0.584 − 0.909i)39-s + (−0.857 + 0.989i)43-s + (−0.654 − 0.755i)48-s + (−0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ -0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142558364\)
\(L(\frac12)\) \(\approx\) \(1.142558364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 - 0.909i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (1.07 - 0.153i)T + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
79 \( 1 + 0.830T + T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
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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

−9.875563484006129846394877730194, −9.535629690657220204248449316956, −8.622471333678032211105711674228, −7.71995704367415177540868879685, −7.30445784425111556398713820261, −5.89786453269836428042016942074, −4.89087427686062687882039265577, −4.05645765900581912639012346045, −3.16806240550113508445465970640, −2.31217379234873120351619679910, 0.995239217340561440223864612329, 2.23673907270495502409446480007, 3.15201608739993493362129910514, 4.77403775589028185917240222198, 5.48595655174089943305834232445, 6.61383519158652286124404246923, 7.00764859182785639170706734614, 8.052798724800147394048408822024, 8.832483310405405601675576374992, 9.833528463903955005566024907963

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