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Properties

Label 2-1184-37.36-c1-0-3
Degree $2$
Conductor $1184$
Sign $-i$
Analytic cond. $9.45428$
Root an. cond. $3.07478$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 7.08i·19-s + 3.08i·23-s + 5·25-s + 11.0i·31-s + 6.08·37-s − 12.1·41-s − 0.917i·43-s − 7·49-s + 12.1·53-s + 15.0i·59-s − 12.1·73-s − 4.91i·79-s + 9·81-s + 12.1·101-s + ⋯
L(s)  = 1  − 9-s + 1.62i·19-s + 0.642i·23-s + 25-s + 1.99i·31-s + 0.999·37-s − 1.89·41-s − 0.139i·43-s − 49-s + 1.67·53-s + 1.96i·59-s − 1.42·73-s − 0.553i·79-s + 81-s + 1.21·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1184\)    =    \(2^{5} \cdot 37\)
Sign: $-i$
Analytic conductor: \(9.45428\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1184} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1184,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.111035542\)
\(L(\frac12)\) \(\approx\) \(1.111035542\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 - 6.08T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 - 3.08iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 11.0iT - 31T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 0.917iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 15.0iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 4.91iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.13133352599580041561726420375, −8.958875311300416068581013197514, −8.468696073369461140220766889723, −7.56227107781882141098698021456, −6.59565657630872792261746898435, −5.72187041611065630535091321503, −4.97603540632587399766801761318, −3.69736995130601668399068878704, −2.86219861786989402823004291934, −1.44266272916152121959030008419, 0.48609899715687249450637051819, 2.31358425436441148415819231153, 3.16404351373553679324922443893, 4.44756499046267891282718274794, 5.27045239912424028216931333029, 6.25553443515884990181237613985, 6.99517392545376136968390546968, 8.060059597852300750454903456426, 8.735886747540492507777203140357, 9.465103129593134319193083608614

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