Properties

Label 2-1183-7.2-c1-0-19
Degree $2$
Conductor $1183$
Sign $-0.915 + 0.403i$
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 + (1.37 + 2.38i)3-s + (−1.85 − 3.20i)4-s + (0.491 − 0.850i)5-s − 6.57·6-s + (1.69 − 2.03i)7-s + 4.06·8-s + (−2.28 + 3.95i)9-s + (1.17 + 2.03i)10-s + (−0.293 − 0.509i)11-s + (5.09 − 8.82i)12-s + (2.17 + 5.93i)14-s + 2.70·15-s + (−1.15 + 1.99i)16-s + (3.22 + 5.58i)17-s + (−5.45 − 9.45i)18-s + ⋯
L(s)  = 1  + (−0.844 + 1.46i)2-s + (0.794 + 1.37i)3-s + (−0.925 − 1.60i)4-s + (0.219 − 0.380i)5-s − 2.68·6-s + (0.640 − 0.767i)7-s + 1.43·8-s + (−0.761 + 1.31i)9-s + (0.370 + 0.642i)10-s + (−0.0886 − 0.153i)11-s + (1.47 − 2.54i)12-s + (0.581 + 1.58i)14-s + 0.697·15-s + (−0.288 + 0.498i)16-s + (0.782 + 1.35i)17-s + (−1.28 − 2.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272682220\)
\(L(\frac12)\) \(\approx\) \(1.272682220\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.69 + 2.03i)T \)
13 \( 1 \)
good2 \( 1 + (1.19 - 2.06i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.37 - 2.38i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.491 + 0.850i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.293 + 0.509i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.22 - 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.91 - 3.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.13 - 7.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + (1.49 + 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.877 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.67T + 41T^{2} \)
43 \( 1 - 6.38T + 43T^{2} \)
47 \( 1 + (2.17 - 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.212 + 0.368i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.00 - 5.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.10 - 1.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.50 - 6.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 + (-2.46 - 4.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 2.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 + (1.04 - 1.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.69T + 97T^{2} \)
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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

−9.931603273933466687194930516636, −9.283430735659107849599234672289, −8.488679842523830593103791053630, −7.973244232174603798253448483109, −7.31436628603161298387569202692, −5.88815083630141567121661269663, −5.38233496988131434761946333193, −4.26498697675235411183452722206, −3.56731546309962100460960668648, −1.54008603065230038616363744320, 0.68418433516236273367379210786, 2.00047283666254806856902486599, 2.43804046918812726096077969034, 3.23440885229328618974170815427, 4.76304515611060771165530975732, 6.20055535482824554517766441355, 7.18172489704164016407464454054, 8.001452961607083991764459399772, 8.592382294485326420182044795860, 9.209461526599998063665252823886

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