Properties

Label 2-1133-1133.373-c0-0-0
Degree $2$
Conductor $1133$
Sign $0.875 + 0.482i$
Analytic cond. $0.565440$
Root an. cond. $0.751957$
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  + (0.538 − 0.100i)3-s + (0.739 − 0.673i)4-s + (−0.0505 + 0.177i)5-s + (−0.653 + 0.253i)9-s + (0.932 − 0.361i)11-s + (0.329 − 0.436i)12-s + (−0.00931 + 0.100i)15-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)20-s + (−0.510 − 0.197i)23-s + (0.821 + 0.508i)25-s + (−0.791 + 0.489i)27-s + (−0.0505 + 0.544i)31-s + (0.465 − 0.288i)33-s + (−0.312 + 0.627i)36-s + (0.726 − 0.961i)37-s + ⋯
L(s)  = 1  + (0.538 − 0.100i)3-s + (0.739 − 0.673i)4-s + (−0.0505 + 0.177i)5-s + (−0.653 + 0.253i)9-s + (0.932 − 0.361i)11-s + (0.329 − 0.436i)12-s + (−0.00931 + 0.100i)15-s + (0.0922 − 0.995i)16-s + (0.0822 + 0.165i)20-s + (−0.510 − 0.197i)23-s + (0.821 + 0.508i)25-s + (−0.791 + 0.489i)27-s + (−0.0505 + 0.544i)31-s + (0.465 − 0.288i)33-s + (−0.312 + 0.627i)36-s + (0.726 − 0.961i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ 0.875 + 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417445904\)
\(L(\frac12)\) \(\approx\) \(1.417445904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.932 + 0.361i)T \)
103 \( 1 + (-0.0922 + 0.995i)T \)
good2 \( 1 + (-0.739 + 0.673i)T^{2} \)
3 \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \)
5 \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \)
7 \( 1 + (0.602 - 0.798i)T^{2} \)
13 \( 1 + (0.602 - 0.798i)T^{2} \)
17 \( 1 + (-0.445 - 0.895i)T^{2} \)
19 \( 1 + (-0.932 - 0.361i)T^{2} \)
23 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
29 \( 1 + (0.850 + 0.526i)T^{2} \)
31 \( 1 + (0.0505 - 0.544i)T + (-0.982 - 0.183i)T^{2} \)
37 \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \)
41 \( 1 + (0.850 - 0.526i)T^{2} \)
43 \( 1 + (0.273 - 0.961i)T^{2} \)
47 \( 1 + 1.70T + T^{2} \)
53 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
59 \( 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2} \)
61 \( 1 + (-0.445 - 0.895i)T^{2} \)
67 \( 1 + (-0.397 + 0.798i)T + (-0.602 - 0.798i)T^{2} \)
71 \( 1 + (-0.538 - 1.89i)T + (-0.850 + 0.526i)T^{2} \)
73 \( 1 + (0.850 - 0.526i)T^{2} \)
79 \( 1 + (0.850 + 0.526i)T^{2} \)
83 \( 1 + (0.602 - 0.798i)T^{2} \)
89 \( 1 + (-0.136 - 0.124i)T + (0.0922 + 0.995i)T^{2} \)
97 \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \)
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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

−9.867064578716767647397673309824, −9.177220924242928359010407750225, −8.325342282957492267552754546504, −7.45199810543239534380303416839, −6.55591430024702658038128151087, −5.89968539557712553770584601016, −4.88073238384484906641109529737, −3.50778782632998707800137130968, −2.63308552110472121140849095050, −1.48142413004300878943283225615, 1.80318024495587317584820183472, 2.94639574426087807163585565396, 3.70136497419226948858814205922, 4.75265080951619087411004083117, 6.18375375084934722796306783986, 6.67373922164017303291023795346, 7.83812090656628306454750958217, 8.311237121754159554869278167927, 9.203397241798701098466104225610, 9.910929060285606833185445313233

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