Properties

Label 1-1157-1157.584-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.436 + 0.899i$
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  + (0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (−0.841 + 0.540i)11-s − 12-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (0.142 − 0.989i)3-s + (−0.142 − 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (−0.841 − 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (−0.841 + 0.540i)11-s − 12-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2722734621 - 0.1706194989i\)
\(L(\frac12)\) \(\approx\) \(-0.2722734621 - 0.1706194989i\)
\(L(1)\) \(\approx\) \(0.6158163628 - 0.7543772201i\)
\(L(1)\) \(\approx\) \(0.6158163628 - 0.7543772201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.142 + 0.989i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−21.82687656561634579086289130965, −21.40925632613941266209824815963, −20.53692423635844345942429816492, −19.90502282310662968423711864710, −18.72185333123918069358023155412, −17.90569935549531115894848893371, −16.78024888077241819543456525379, −16.55235781555461845768377009317, −15.44758828386098589154405505537, −14.914997512257904392971860905476, −14.580032060331316339725537881034, −13.49759287305802718680791054841, −12.75889161947330606561160318753, −11.39251136630884412911522501769, −11.08469616477051487770543414017, −10.29113798900282548898984981165, −8.79604935035073684539369978841, −8.28876927051288160545709063871, −7.56515350345959097991136539855, −6.598445719962773887395963628944, −5.531115865752652706815707916, −4.71768253190321310896232686925, −4.01478568034134719783337476433, −3.35750105645236190864618643391, −2.3171702267114514529599562138, 0.09729348281250066620658577234, 1.39711913524403056525663431510, 2.240393631546382870543526969652, 3.024440588029810936586183076826, 4.32328971510001403498641149548, 5.001324937584884268495731860285, 5.79962111145624526552974460090, 7.02580858280340400453951047618, 7.75890469743229770728462226970, 8.72410976142826054858798462082, 9.33709016648591719329848209241, 10.943457624496465988941258647614, 11.27952061572650051719124114638, 12.121295903648681011267820052357, 12.82444412213500532402187444144, 13.244296610895233473049712685049, 14.32526256969585965621314965184, 15.11978422962927703819828603198, 15.5619636930531612851285987075, 16.93699834943978507075545022950, 17.90848921613656839263254537044, 18.54238645044952337892111874412, 19.15773350931364434160114217832, 20.017471411604273060741094763911, 20.549680117263876984483065011831

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