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Properties

Label 1-1183-1183.1137-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.998 + 0.0467i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.464 − 0.885i)2-s + (−0.278 + 0.960i)3-s + (−0.568 + 0.822i)4-s + (0.774 − 0.632i)5-s + (0.979 − 0.200i)6-s + (0.992 + 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (−0.534 − 0.845i)11-s + (−0.632 − 0.774i)12-s + (0.391 + 0.919i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (−0.0804 + 0.996i)18-s + (0.866 − 0.5i)19-s + (0.0804 + 0.996i)20-s + ⋯
L(s)  = 1  + (−0.464 − 0.885i)2-s + (−0.278 + 0.960i)3-s + (−0.568 + 0.822i)4-s + (0.774 − 0.632i)5-s + (0.979 − 0.200i)6-s + (0.992 + 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (−0.534 − 0.845i)11-s + (−0.632 − 0.774i)12-s + (0.391 + 0.919i)15-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (−0.0804 + 0.996i)18-s + (0.866 − 0.5i)19-s + (0.0804 + 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.998 + 0.0467i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.998 + 0.0467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008560758669 - 0.3661706542i\)
\(L(\frac12)\) \(\approx\) \(0.008560758669 - 0.3661706542i\)
\(L(1)\) \(\approx\) \(0.6130143598 - 0.2265712312i\)
\(L(1)\) \(\approx\) \(0.6130143598 - 0.2265712312i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.464 - 0.885i)T \)
3 \( 1 + (-0.278 + 0.960i)T \)
5 \( 1 + (0.774 - 0.632i)T \)
11 \( 1 + (-0.534 - 0.845i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.845 - 0.534i)T \)
31 \( 1 + (-0.316 + 0.948i)T \)
37 \( 1 + (-0.663 - 0.748i)T \)
41 \( 1 + (-0.721 + 0.692i)T \)
43 \( 1 + (-0.948 + 0.316i)T \)
47 \( 1 + (0.0804 + 0.996i)T \)
53 \( 1 + (-0.919 + 0.391i)T \)
59 \( 1 + (0.935 + 0.354i)T \)
61 \( 1 + (-0.799 + 0.600i)T \)
67 \( 1 + (0.0804 + 0.996i)T \)
71 \( 1 + (-0.960 - 0.278i)T \)
73 \( 1 + (-0.999 + 0.0402i)T \)
79 \( 1 + (-0.996 + 0.0804i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.160 + 0.987i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.13312435814845203204669882930, −20.626238028085497274024728827377, −19.89398338275554031735420130223, −18.751344556213640350271825923746, −18.52121037615373626645488285399, −17.74796277471625827629052856936, −17.189933966089187738584407084908, −16.472245402744937699951018478893, −15.32334652613187231631475138003, −14.66788995105560685095509803364, −13.87408808909962746400124012170, −13.26877417072220201055131152657, −12.42655015687848026771250403050, −11.31472941483999941617323972161, −10.299701960178001677894438725251, −9.86929704037785028140725122923, −8.69506873664791923796322427555, −7.80789056546139645731141314657, −7.207829325482873631936658846, −6.42721066728602090672644240563, −5.70439828701379029031551956461, −5.06875462152576648683187404127, −3.537030410629936774630857771213, −2.02922943706526410596714129735, −1.597066826239936358592276930110, 0.17688898651840821273296673182, 1.39662976018098850989152107822, 2.67419483124521149931627201282, 3.36331984111499156777278377772, 4.50690585694633159919444994100, 5.20160258601895814568227429538, 5.96538090617995963153254065476, 7.44968667838270682028252734945, 8.539972869943526619638041821075, 9.106407664719067066187712006764, 9.8558637852835008599502560257, 10.40442962281694695967810053994, 11.38315088820176721599059778955, 11.897656730361915343856471822081, 12.98382739025658321855729875512, 13.725764661946135005241181117951, 14.36790956367489450944635471779, 15.95503501621391844572142277873, 16.17921508303724364178455976560, 17.04786027078554895078101844412, 17.84035222927758237332129875979, 18.34088246453571821274786825891, 19.48890542892396103613587909646, 20.42481112583837076458898781362, 20.66937313142882956786307064033

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