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Properties

Label 2-1104-69.68-c1-0-12
Degree $2$
Conductor $1104$
Sign $-0.213 - 0.977i$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
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  + (−1.28 + 1.16i)3-s − 0.936·5-s + 3.88i·7-s + (0.280 − 2.98i)9-s + 4.27·11-s + 2.56·13-s + (1.19 − 1.09i)15-s + 7.60·17-s − 6.07i·19-s + (−4.53 − 4.98i)21-s + (−2.39 + 4.15i)23-s − 4.12·25-s + (3.12 + 4.15i)27-s + 5.97i·29-s − 3.68·31-s + ⋯
L(s)  = 1  + (−0.739 + 0.673i)3-s − 0.418·5-s + 1.46i·7-s + (0.0935 − 0.995i)9-s + 1.28·11-s + 0.710·13-s + (0.309 − 0.281i)15-s + 1.84·17-s − 1.39i·19-s + (−0.989 − 1.08i)21-s + (−0.500 + 0.865i)23-s − 0.824·25-s + (0.601 + 0.799i)27-s + 1.10i·29-s − 0.661·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-0.213 - 0.977i$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ -0.213 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.211260732\)
\(L(\frac12)\) \(\approx\) \(1.211260732\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.28 - 1.16i)T \)
23 \( 1 + (2.39 - 4.15i)T \)
good5 \( 1 + 0.936T + 5T^{2} \)
7 \( 1 - 3.88iT - 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 - 7.60T + 17T^{2} \)
19 \( 1 + 6.07iT - 19T^{2} \)
29 \( 1 - 5.97iT - 29T^{2} \)
31 \( 1 + 3.68T + 31T^{2} \)
37 \( 1 + 2.18iT - 37T^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 - 6.07iT - 43T^{2} \)
47 \( 1 - 1.30iT - 47T^{2} \)
53 \( 1 - 8.13T + 53T^{2} \)
59 \( 1 + 4.66iT - 59T^{2} \)
61 \( 1 - 7.77iT - 61T^{2} \)
67 \( 1 + 11.6iT - 67T^{2} \)
71 \( 1 - 2.33iT - 71T^{2} \)
73 \( 1 + 5.68T + 73T^{2} \)
79 \( 1 - 1.70iT - 79T^{2} \)
83 \( 1 - 0.525T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 - 9.96iT - 97T^{2} \)
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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

−9.925883295733480607569130887913, −9.290432177657733741871531783051, −8.727856260384349520062001438522, −7.60516667036970005651597808548, −6.45323711302654802367521590844, −5.78411070336412348091723216800, −5.07169712216329259340619094788, −3.89920686394283528915318382886, −3.11941203863755922339687741172, −1.32505437960066618784639624737, 0.70436815558550491368671655378, 1.66507272036842401040708134867, 3.75844864476652824854209092933, 4.05337093248789442059553188135, 5.58418188907986212848122828523, 6.24436898930483485866271692002, 7.22193576556034279365193057267, 7.71354893288426491043496024267, 8.551173632282228858521182136971, 10.03794981287197369476835225981

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