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Properties

Label 2-1183-7.4-c1-0-46
Degree $2$
Conductor $1183$
Sign $0.989 - 0.145i$
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  + (0.545 + 0.945i)2-s + (−0.697 + 1.20i)3-s + (0.403 − 0.699i)4-s + (−0.813 − 1.40i)5-s − 1.52·6-s + (−2.51 − 0.817i)7-s + 3.06·8-s + (0.526 + 0.911i)9-s + (0.888 − 1.53i)10-s + (−0.646 + 1.12i)11-s + (0.563 + 0.975i)12-s + (−0.600 − 2.82i)14-s + 2.26·15-s + (0.866 + 1.50i)16-s + (2.70 − 4.68i)17-s + (−0.574 + 0.995i)18-s + ⋯
L(s)  = 1  + (0.386 + 0.668i)2-s + (−0.402 + 0.697i)3-s + (0.201 − 0.349i)4-s + (−0.363 − 0.629i)5-s − 0.622·6-s + (−0.951 − 0.309i)7-s + 1.08·8-s + (0.175 + 0.303i)9-s + (0.280 − 0.486i)10-s + (−0.195 + 0.337i)11-s + (0.162 + 0.281i)12-s + (−0.160 − 0.755i)14-s + 0.586·15-s + (0.216 + 0.375i)16-s + (0.656 − 1.13i)17-s + (−0.135 + 0.234i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.989 - 0.145i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.669081972\)
\(L(\frac12)\) \(\approx\) \(1.669081972\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.51 + 0.817i)T \)
13 \( 1 \)
good2 \( 1 + (-0.545 - 0.945i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.697 - 1.20i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.813 + 1.40i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.646 - 1.12i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.70 + 4.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.755 + 1.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.32 - 2.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.81T + 29T^{2} \)
31 \( 1 + (-3.64 + 6.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.47 + 6.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (-3.58 - 6.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.33 - 4.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.386 + 0.670i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.18 + 5.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (-3.60 + 6.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.88 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.42T + 83T^{2} \)
89 \( 1 + (0.833 + 1.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.755516898940745064178142303051, −9.228409087098510900346208885443, −7.77917178691815484180223799449, −7.32421586020800271871235778683, −6.27341580248281002025117579056, −5.50709141301257052771421227365, −4.66319594482855149261670847778, −4.14794784714080237981857849960, −2.62428352089150039287902621983, −0.78385107648014347510512632545, 1.20946160437068909447996333427, 2.65793214293305008676463473026, 3.35491599375591071788956891812, 4.21181231665348454247835955904, 5.68413076067427680977666341754, 6.54106144027471289172590520339, 7.07909663449170402122652111249, 7.968462656515553941233356847876, 8.876590507067298349548104072896, 10.19292116682649089037973120642

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