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Properties

Label 2-1183-7.4-c1-0-10
Degree $2$
Conductor $1183$
Sign $0.942 - 0.333i$
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  + (−0.929 − 1.60i)2-s + (−1.14 + 1.98i)3-s + (−0.726 + 1.25i)4-s + (0.0986 + 0.170i)5-s + 4.26·6-s + (−2.62 − 0.317i)7-s − 1.01·8-s + (−1.13 − 1.95i)9-s + (0.183 − 0.317i)10-s + (2.09 − 3.62i)11-s + (−1.66 − 2.88i)12-s + (1.92 + 4.52i)14-s − 0.452·15-s + (2.39 + 4.15i)16-s + (−0.420 + 0.728i)17-s + (−2.10 + 3.64i)18-s + ⋯
L(s)  = 1  + (−0.656 − 1.13i)2-s + (−0.662 + 1.14i)3-s + (−0.363 + 0.629i)4-s + (0.0441 + 0.0764i)5-s + 1.74·6-s + (−0.992 − 0.120i)7-s − 0.359·8-s + (−0.377 − 0.653i)9-s + (0.0579 − 0.100i)10-s + (0.630 − 1.09i)11-s + (−0.481 − 0.833i)12-s + (0.515 + 1.20i)14-s − 0.116·15-s + (0.599 + 1.03i)16-s + (−0.102 + 0.176i)17-s + (−0.495 + 0.858i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.942 - 0.333i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5809907178\)
\(L(\frac12)\) \(\approx\) \(0.5809907178\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 + 0.317i)T \)
13 \( 1 \)
good2 \( 1 + (0.929 + 1.60i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.14 - 1.98i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.0986 - 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.420 - 0.728i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.675 + 1.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 + (-0.640 + 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.52 + 2.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.39T + 41T^{2} \)
43 \( 1 - 5.32T + 43T^{2} \)
47 \( 1 + (-5.83 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.02 - 5.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.68 - 9.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.97T + 71T^{2} \)
73 \( 1 + (1.94 - 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.07T + 83T^{2} \)
89 \( 1 + (-5.99 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 19.4T + 97T^{2} \)
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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

−9.964136802780453926845820667015, −9.149772725126343681125711395651, −8.930306147885224328690722674582, −7.42295330728234404272441131558, −6.09959163801645596728178638814, −5.74556707823868147521878219921, −4.23581636877354541559655881652, −3.57318582637431531353791531278, −2.61421934187211140168286093510, −0.882986977365847155704218535426, 0.46661322384760203045745736940, 1.98717490099525154133052795015, 3.50035290009925227699128695380, 5.03217409105379802858790112584, 6.05780457287202048825481807390, 6.51793768785482343778719795484, 7.23547559441308783644681948848, 7.62354764912507265483983139908, 8.949725823052016050644158775987, 9.320166127519569169114693655444

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