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Properties

Label 1-1157-1157.10-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.307 + 0.951i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.981 + 0.189i)2-s + (−0.371 + 0.928i)3-s + (0.928 − 0.371i)4-s + (0.841 − 0.540i)5-s + (0.189 − 0.981i)6-s + (−0.458 + 0.888i)7-s + (−0.841 + 0.540i)8-s + (−0.723 − 0.690i)9-s + (−0.723 + 0.690i)10-s + (0.888 − 0.458i)11-s + i·12-s + (0.281 − 0.959i)14-s + (0.189 + 0.981i)15-s + (0.723 − 0.690i)16-s + (−0.981 − 0.189i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (−0.371 + 0.928i)3-s + (0.928 − 0.371i)4-s + (0.841 − 0.540i)5-s + (0.189 − 0.981i)6-s + (−0.458 + 0.888i)7-s + (−0.841 + 0.540i)8-s + (−0.723 − 0.690i)9-s + (−0.723 + 0.690i)10-s + (0.888 − 0.458i)11-s + i·12-s + (0.281 − 0.959i)14-s + (0.189 + 0.981i)15-s + (0.723 − 0.690i)16-s + (−0.981 − 0.189i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7294663501 + 0.5310569205i\)
\(L(\frac12)\) \(\approx\) \(0.7294663501 + 0.5310569205i\)
\(L(1)\) \(\approx\) \(0.6699478527 + 0.2456954831i\)
\(L(1)\) \(\approx\) \(0.6699478527 + 0.2456954831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
3 \( 1 + (-0.371 + 0.928i)T \)
5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.458 + 0.888i)T \)
11 \( 1 + (0.888 - 0.458i)T \)
17 \( 1 + (-0.981 - 0.189i)T \)
19 \( 1 + (-0.690 + 0.723i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (0.998 - 0.0475i)T \)
31 \( 1 + (0.281 - 0.959i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.618 + 0.786i)T \)
43 \( 1 + (0.998 + 0.0475i)T \)
47 \( 1 + (-0.142 + 0.989i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.371 + 0.928i)T \)
61 \( 1 + (0.814 + 0.580i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.0475 - 0.998i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
97 \( 1 + (0.888 + 0.458i)T \)
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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

−20.94399041308376000938431858568, −20.01738837394589241104912306271, −19.41056364733592488087023356770, −18.898536699263009246248388181062, −17.78503589392817317335150255084, −17.420120351737800771781357144725, −17.034363077078689948032962504261, −15.9904540370311153872956654662, −14.900006893030884287262448879514, −13.96303152420292890231106597131, −13.18830252317072704823854425564, −12.48369660491898208866549032873, −11.493621423434341950847571501019, −10.71881645344725735922873828011, −10.21933608344225433552030798149, −9.13141069398584211415029792597, −8.47669862707182471795381975498, −7.1375660167083339061889137084, −6.74120053816155491513334115854, −6.40326320647370318714860601371, −4.95623735598011974446790065451, −3.494980531724856916910307383435, −2.454876297445626851748610163379, −1.70217020930755692476749335903, −0.6964341518398601211909982930, 0.89782833975203758773886010138, 2.151003887659854583965948190474, 3.06342861813809398651610223972, 4.391606276372748127646060106923, 5.452808243587719129212099091079, 6.146859645503431342987580321829, 6.63892178962396867063003054521, 8.28807637610817404103738831157, 9.063558061058186647592978848913, 9.26796628954281166641313219515, 10.192162647536219018215509616012, 10.968649191359356328463091773417, 11.82822390256296649376138083527, 12.54153086708493485454013143851, 13.800181457327234409151378352239, 14.79177880356418365414115321447, 15.43336708734860044275458761590, 16.25117048028328205894243634711, 16.80004246819897820582851520168, 17.45779025964455907235481506187, 18.11415321583365657603886022338, 19.17566228887921202114229747377, 19.7608974703455948245887234549, 20.87023238058782207358304610183, 21.18818148084990602982533288936

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