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Properties

Label 2-1183-7.4-c1-0-43
Degree $2$
Conductor $1183$
Sign $-0.992 + 0.119i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  + (−1.19 − 2.06i)2-s + (−0.352 + 0.610i)3-s + (−1.84 + 3.19i)4-s + (0.384 + 0.666i)5-s + 1.68·6-s + (−2.49 − 0.882i)7-s + 4.02·8-s + (1.25 + 2.16i)9-s + (0.917 − 1.58i)10-s + (−0.556 + 0.963i)11-s + (−1.30 − 2.25i)12-s + (1.15 + 6.20i)14-s − 0.543·15-s + (−1.11 − 1.92i)16-s + (−3.52 + 6.11i)17-s + (2.98 − 5.16i)18-s + ⋯
L(s)  = 1  + (−0.843 − 1.46i)2-s + (−0.203 + 0.352i)3-s + (−0.921 + 1.59i)4-s + (0.172 + 0.298i)5-s + 0.686·6-s + (−0.942 − 0.333i)7-s + 1.42·8-s + (0.417 + 0.722i)9-s + (0.290 − 0.502i)10-s + (−0.167 + 0.290i)11-s + (−0.375 − 0.650i)12-s + (0.307 + 1.65i)14-s − 0.140·15-s + (−0.277 − 0.481i)16-s + (−0.855 + 1.48i)17-s + (0.703 − 1.21i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.992 + 0.119i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.992 + 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3597924852\)
\(L(\frac12)\) \(\approx\) \(0.3597924852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.49 + 0.882i)T \)
13 \( 1 \)
good2 \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.352 - 0.610i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.384 - 0.666i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.556 - 0.963i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.52 - 6.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.86 + 3.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.01 + 5.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.31T + 29T^{2} \)
31 \( 1 + (-1.26 + 2.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0519 - 0.0899i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.12T + 41T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 + (4.69 + 8.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.73 + 9.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.54 + 7.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.69 + 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.79 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.13T + 71T^{2} \)
73 \( 1 + (3.83 - 6.64i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.75 + 4.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.42T + 83T^{2} \)
89 \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−9.901309507906728600751979752551, −8.718658007357837141240875503379, −8.205129100383182245208753042901, −6.93322615086565636179553686090, −6.15404582249758085922127746722, −4.55655464655565313869860679071, −3.94164595077683812610488564078, −2.71425521132808910222729595449, −1.96193147049901540754874251846, −0.24379413143037134384970796972, 1.10395317934947109268322290622, 2.96399519953680450647706790927, 4.43406695898783820337307490067, 5.64902812830913920736262502792, 6.11270543708180026186400541194, 6.96130320721264900530218190445, 7.46960249163606558830540765592, 8.538007943355210466264311495733, 9.323218278383448341473015361174, 9.568909513302087125978832383045

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