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Properties

Label 2-1183-7.2-c1-0-46
Degree $2$
Conductor $1183$
Sign $0.632 - 0.774i$
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  + (−0.438 + 0.759i)2-s + (0.377 + 0.653i)3-s + (0.615 + 1.06i)4-s + (0.132 − 0.230i)5-s − 0.662·6-s + (0.588 − 2.57i)7-s − 2.83·8-s + (1.21 − 2.10i)9-s + (0.116 + 0.201i)10-s + (−0.227 − 0.394i)11-s + (−0.464 + 0.804i)12-s + (1.70 + 1.57i)14-s + 0.200·15-s + (0.0124 − 0.0214i)16-s + (1.82 + 3.16i)17-s + (1.06 + 1.84i)18-s + ⋯
L(s)  = 1  + (−0.310 + 0.537i)2-s + (0.217 + 0.377i)3-s + (0.307 + 0.532i)4-s + (0.0594 − 0.102i)5-s − 0.270·6-s + (0.222 − 0.974i)7-s − 1.00·8-s + (0.404 − 0.701i)9-s + (0.0368 + 0.0638i)10-s + (−0.0686 − 0.118i)11-s + (−0.134 + 0.232i)12-s + (0.454 + 0.421i)14-s + 0.0518·15-s + (0.00310 − 0.00536i)16-s + (0.443 + 0.768i)17-s + (0.251 + 0.435i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.632 - 0.774i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.746856162\)
\(L(\frac12)\) \(\approx\) \(1.746856162\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.588 + 2.57i)T \)
13 \( 1 \)
good2 \( 1 + (0.438 - 0.759i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.377 - 0.653i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.132 + 0.230i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.227 + 0.394i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.77 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.53 + 2.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.612T + 29T^{2} \)
31 \( 1 + (-5.35 - 9.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.13T + 41T^{2} \)
43 \( 1 - 6.76T + 43T^{2} \)
47 \( 1 + (-3.73 + 6.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.89 - 6.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.81 + 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.797 + 1.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.89T + 71T^{2} \)
73 \( 1 + (1.42 + 2.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.36 - 7.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.51T + 83T^{2} \)
89 \( 1 + (4.21 - 7.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.61T + 97T^{2} \)
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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

−9.736682514609530941161720937719, −8.917306277844026881950965653748, −8.296181419149084403302679067421, −7.25213880425291654583917942868, −6.90390864155713312023116444293, −5.84523039217784119159669883278, −4.59849386189648384795238453195, −3.71674008976046642724287042984, −2.87795208436130276161484370237, −1.05388503968016653335350480488, 1.13043645442968998443950285627, 2.28593727097766544896555531961, 2.85556648522589223535717794078, 4.51710560023869350828235288173, 5.55831167393108159045340501268, 6.17363849911307990888714963174, 7.34126980678782970578716683187, 8.009334179082938534683016790000, 9.031881616703878174994855903855, 9.699707174808721321048742598493

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