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Properties

Label 1-1157-1157.305-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.254 - 0.967i$
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  + (−0.371 − 0.928i)2-s + (0.0237 + 0.999i)3-s + (−0.723 + 0.690i)4-s + (−0.415 + 0.909i)5-s + (0.919 − 0.393i)6-s + (−0.986 − 0.165i)7-s + (0.909 + 0.415i)8-s + (−0.998 + 0.0475i)9-s + (0.998 + 0.0475i)10-s + (−0.814 − 0.580i)11-s + (−0.707 − 0.707i)12-s + (0.212 + 0.977i)14-s + (−0.919 − 0.393i)15-s + (0.0475 − 0.998i)16-s + (−0.371 + 0.928i)17-s + (0.415 + 0.909i)18-s + ⋯
L(s)  = 1  + (−0.371 − 0.928i)2-s + (0.0237 + 0.999i)3-s + (−0.723 + 0.690i)4-s + (−0.415 + 0.909i)5-s + (0.919 − 0.393i)6-s + (−0.986 − 0.165i)7-s + (0.909 + 0.415i)8-s + (−0.998 + 0.0475i)9-s + (0.998 + 0.0475i)10-s + (−0.814 − 0.580i)11-s + (−0.707 − 0.707i)12-s + (0.212 + 0.977i)14-s + (−0.919 − 0.393i)15-s + (0.0475 − 0.998i)16-s + (−0.371 + 0.928i)17-s + (0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.254 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1950744342 - 0.1503358171i\)
\(L(\frac12)\) \(\approx\) \(0.1950744342 - 0.1503358171i\)
\(L(1)\) \(\approx\) \(0.5170812494 + 0.04207630017i\)
\(L(1)\) \(\approx\) \(0.5170812494 + 0.04207630017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.371 - 0.928i)T \)
3 \( 1 + (0.0237 + 0.999i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.986 - 0.165i)T \)
11 \( 1 + (-0.814 - 0.580i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (-0.672 + 0.739i)T \)
23 \( 1 + (0.304 + 0.952i)T \)
29 \( 1 + (0.636 - 0.771i)T \)
31 \( 1 + (-0.212 - 0.977i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.853 - 0.520i)T \)
43 \( 1 + (-0.636 - 0.771i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.999 - 0.0237i)T \)
61 \( 1 + (-0.436 + 0.899i)T \)
67 \( 1 + (0.690 - 0.723i)T \)
71 \( 1 + (0.995 + 0.0950i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (0.800 + 0.599i)T \)
97 \( 1 + (0.814 - 0.580i)T \)
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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

−21.51317939077126084642071847845, −20.15529317610242562035610988113, −19.832439716291742384714594258349, −19.00141374153789389827064633927, −18.25602005267027975384697318715, −17.648315164546160180565806476718, −16.721050568432222348740070907772, −16.10901841246063064056124049206, −15.46698413369439304295536644523, −14.49303469709756284377067201999, −13.51062362104988830058331706814, −12.84332692276318175824831142978, −12.49764625614932801245190194710, −11.23846336322050483917254170103, −10.140307907323821305749286440351, −9.11868077976521009428445009907, −8.625729575752623508665679516742, −7.78355317384224317866818778498, −6.90252194279947607240988223834, −6.4595998651441869795091760308, −5.20064093976357065797261150864, −4.74388295671526600491473377007, −3.22169119311756769803749308728, −2.03719133796970094837427330332, −0.73477717626763488733028390398, 0.1666113945113145763324933490, 2.126244513321037963848648831541, 3.10073352237056928979809529335, 3.57348463214463621561609598570, 4.33879405301919906105239561407, 5.60535781439238040805483248543, 6.54850356633086420150687032192, 7.85105684692801104562702192998, 8.46206506704483535272033915072, 9.5126542373952564662272743570, 10.26698088091813908233625453627, 10.61721263051977348050372543525, 11.42440744523684919251450230059, 12.26012067562636392486259071409, 13.36346029168216156034429364952, 13.891032221617588703850493926718, 15.177895120492616229740533304, 15.547379002967161519651606008104, 16.68986786606934603168592611820, 17.12287163683234566434891235279, 18.31090587246081278312481496427, 19.0122785669932565701052412384, 19.5444755426309583047576157724, 20.2827095611698191261395588963, 21.27307023207833363844611649143

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