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Properties

Label 1-1157-1157.110-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.964 - 0.265i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (−0.458 + 0.888i)2-s + (−0.814 − 0.580i)3-s + (−0.580 − 0.814i)4-s + (0.989 − 0.142i)5-s + (0.888 − 0.458i)6-s + (0.928 + 0.371i)7-s + (0.989 − 0.142i)8-s + (0.327 + 0.945i)9-s + (−0.327 + 0.945i)10-s + (0.371 + 0.928i)11-s + i·12-s + (−0.755 + 0.654i)14-s + (−0.888 − 0.458i)15-s + (−0.327 + 0.945i)16-s + (−0.888 + 0.458i)17-s + (−0.989 − 0.142i)18-s + ⋯
L(s)  = 1  + (−0.458 + 0.888i)2-s + (−0.814 − 0.580i)3-s + (−0.580 − 0.814i)4-s + (0.989 − 0.142i)5-s + (0.888 − 0.458i)6-s + (0.928 + 0.371i)7-s + (0.989 − 0.142i)8-s + (0.327 + 0.945i)9-s + (−0.327 + 0.945i)10-s + (0.371 + 0.928i)11-s + i·12-s + (−0.755 + 0.654i)14-s + (−0.888 − 0.458i)15-s + (−0.327 + 0.945i)16-s + (−0.888 + 0.458i)17-s + (−0.989 − 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389678522 - 0.1880374037i\)
\(L(\frac12)\) \(\approx\) \(1.389678522 - 0.1880374037i\)
\(L(1)\) \(\approx\) \(0.8177813747 + 0.1510724809i\)
\(L(1)\) \(\approx\) \(0.8177813747 + 0.1510724809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.458 + 0.888i)T \)
3 \( 1 + (-0.814 - 0.580i)T \)
5 \( 1 + (0.989 - 0.142i)T \)
7 \( 1 + (0.928 + 0.371i)T \)
11 \( 1 + (0.371 + 0.928i)T \)
17 \( 1 + (-0.888 + 0.458i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (-0.189 + 0.981i)T \)
29 \( 1 + (-0.618 - 0.786i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (0.618 - 0.786i)T \)
47 \( 1 + (-0.909 - 0.415i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (-0.690 - 0.723i)T \)
67 \( 1 + (-0.814 - 0.580i)T \)
71 \( 1 + (0.618 - 0.786i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.371 - 0.928i)T \)
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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

−21.203959743317171250569630850101, −20.57229962253072820857175036953, −19.73696344609933338828541509488, −18.4131126185458016339846851566, −18.146259383189824499563224447779, −17.37991684914499373776412189924, −16.66985269479043957015844774892, −16.186864571372305804046528410210, −14.56164800615172983592805839833, −14.189056218464823707926092072077, −13.084834006450668450222566287285, −12.34994500650969700469364249439, −11.28666903695466957174847192203, −10.86702369651936244407884185932, −10.32689287026353677655551756658, −9.24967839740280206134603880095, −8.80359639338693984398453126171, −7.58686875854439813516215051270, −6.46480941241504913151746548158, −5.5726910125765079164601312891, −4.64259168149728236265571720180, −3.91625089932182421627513378507, −2.75667675817292913263157608418, −1.61946484017720104100889324217, −0.84648613384196482568225593317, 0.46262837039814417582240710749, 1.79754636152378217290650354021, 1.983913266887568711572599177974, 4.41654549441282666115385771866, 4.977223346906096494510880862423, 5.84069691770543558320556133485, 6.453133572794798085076672027323, 7.31344309222323069828905404030, 8.084877539929808407914958148581, 9.13198383240575642625562510147, 9.73959105588884394106971904558, 10.83590282682656475718055322391, 11.40320393659977154534532599846, 12.6385724054775731684292536176, 13.319378203407118480665515400554, 14.0392320530881255342163343996, 15.02845339327275756203440228101, 15.571594828957673931158146362961, 16.91071966011330176431744371008, 17.18907972298242777523652488078, 17.91512577933580492765311694834, 18.20246078636353194334307897362, 19.27033817968988995551641422627, 20.099055714648211228834936718213, 21.23639920043352061888957897396

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