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Properties

Label 1-1157-1157.684-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.999 + 0.0167i$
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  + (0.281 − 0.959i)2-s + (0.212 − 0.977i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.877 − 0.479i)6-s + (−0.0713 + 0.997i)7-s + (−0.755 + 0.654i)8-s + (−0.909 − 0.415i)9-s + (0.909 − 0.415i)10-s + (−0.755 − 0.654i)11-s + (−0.707 + 0.707i)12-s + (0.936 + 0.349i)14-s + (0.877 − 0.479i)15-s + (0.415 + 0.909i)16-s + (0.281 + 0.959i)17-s + (−0.654 + 0.755i)18-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)2-s + (0.212 − 0.977i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.877 − 0.479i)6-s + (−0.0713 + 0.997i)7-s + (−0.755 + 0.654i)8-s + (−0.909 − 0.415i)9-s + (0.909 − 0.415i)10-s + (−0.755 − 0.654i)11-s + (−0.707 + 0.707i)12-s + (0.936 + 0.349i)14-s + (0.877 − 0.479i)15-s + (0.415 + 0.909i)16-s + (0.281 + 0.959i)17-s + (−0.654 + 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.999 + 0.0167i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.999 + 0.0167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187393900 + 0.009954619395i\)
\(L(\frac12)\) \(\approx\) \(1.187393900 + 0.009954619395i\)
\(L(1)\) \(\approx\) \(0.9717820170 - 0.4689285894i\)
\(L(1)\) \(\approx\) \(0.9717820170 - 0.4689285894i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.281 - 0.959i)T \)
3 \( 1 + (0.212 - 0.977i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.0713 + 0.997i)T \)
11 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.349 - 0.936i)T \)
23 \( 1 + (0.349 + 0.936i)T \)
29 \( 1 + (-0.0713 + 0.997i)T \)
31 \( 1 + (-0.936 - 0.349i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.212 - 0.977i)T \)
43 \( 1 + (0.0713 + 0.997i)T \)
47 \( 1 + (0.841 + 0.540i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.977 + 0.212i)T \)
61 \( 1 + (0.800 + 0.599i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (0.909 - 0.415i)T \)
83 \( 1 + (0.877 + 0.479i)T \)
97 \( 1 + (0.755 - 0.654i)T \)
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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

−21.187101277039406623335823092803, −20.6823197545877049812042778886, −20.16740377588675281898438833957, −18.80186830829450008486932338970, −17.83237239204836098924188653066, −17.091636549723386600004235423747, −16.52734870310061749618431279588, −16.0326665099115080852873403687, −15.10311447103505314457370746045, −14.27892639103428123874429608868, −13.73345940869792051544990189650, −12.94021065869431310105191258429, −12.122495463486927312919216776855, −10.67916318975915239471644940293, −9.97073424439125468157682055261, −9.38078890716002899239000660865, −8.44736625421781867876274352414, −7.72880962884536602334387202580, −6.746797900443496580243369746227, −5.620219873067409047633254577380, −5.00828336428777787861051835707, −4.32014846816295077123005476912, −3.49426486263689743639350291156, −2.21996857190250093540533842375, −0.42141318358909617899216913715, 1.32158589817787848938880503439, 2.1970992548401851259874123319, 2.84150930323108123465981063170, 3.53877505884172030041106992813, 5.283045428317814368678986018576, 5.73862232285649674558109884616, 6.58708777164248837319246917690, 7.73775971590449448030289481187, 8.78867833106932066774683318724, 9.255987398962267594632963699934, 10.49872300009672482508864984970, 11.07448316910115906993516412751, 11.91027737000917524190701466828, 12.82388521333735649559218047822, 13.25402128690121103364346553482, 14.04771234056620567630385957613, 14.81985175719520482133886773944, 15.466280318277570926721014429, 17.077865466252125680044764341294, 17.83127003270440800015737187166, 18.408572741428423987899276207331, 19.0301045570129851105867860014, 19.45684099740822631146315848353, 20.55486502598304831332918099682, 21.43895129233930020069876945073

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