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Properties

Label 2-1191-1191.35-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.933 + 0.357i$
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  + (−0.5 + 0.866i)3-s − 4-s + (−1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (1.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (1.5 − 0.866i)21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 + 0.866i)28-s − 31-s + (0.499 + 0.866i)36-s + (1 − 1.73i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s − 4-s + (−1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (1.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s + (1.5 − 0.866i)21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 + 0.866i)28-s − 31-s + (0.499 + 0.866i)36-s + (1 − 1.73i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.933 + 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5546120101\)
\(L(\frac12)\) \(\approx\) \(0.5546120101\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
397 \( 1 - T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.719650491537961470731312083948, −9.281008776609150354845338294198, −8.698895641271596281357618521882, −7.32506943170227492889266914057, −6.35862807686730366278487955415, −5.74841708359112710365280960320, −4.48797240556585259461349637436, −3.93147918738295870543552061143, −3.18975106289151085018506111104, −0.66669576585410825697822313884, 1.15323466779719063038657096547, 2.93245060061195969706874171785, 3.70031432019657060922039545343, 5.23803889531485341861914378000, 5.86500816419121904623155290348, 6.41477002112533212718053879528, 7.61901046915063473830617132673, 8.423800458865870935332890082360, 9.112086111019762132601855847828, 9.910804776469886212801322055978

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