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Properties

Label 1-1157-1157.449-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.999 + 0.00656i$
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  + (−0.0950 − 0.995i)2-s + (0.981 + 0.189i)3-s + (−0.981 + 0.189i)4-s + (−0.281 − 0.959i)5-s + (0.0950 − 0.995i)6-s + (−0.971 − 0.235i)7-s + (0.281 + 0.959i)8-s + (0.928 + 0.371i)9-s + (−0.928 + 0.371i)10-s + (0.971 − 0.235i)11-s − 12-s + (−0.142 + 0.989i)14-s + (−0.0950 − 0.995i)15-s + (0.928 − 0.371i)16-s + (0.995 + 0.0950i)17-s + (0.281 − 0.959i)18-s + ⋯
L(s)  = 1  + (−0.0950 − 0.995i)2-s + (0.981 + 0.189i)3-s + (−0.981 + 0.189i)4-s + (−0.281 − 0.959i)5-s + (0.0950 − 0.995i)6-s + (−0.971 − 0.235i)7-s + (0.281 + 0.959i)8-s + (0.928 + 0.371i)9-s + (−0.928 + 0.371i)10-s + (0.971 − 0.235i)11-s − 12-s + (−0.142 + 0.989i)14-s + (−0.0950 − 0.995i)15-s + (0.928 − 0.371i)16-s + (0.995 + 0.0950i)17-s + (0.281 − 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.999 + 0.00656i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.999 + 0.00656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.786872710 + 0.005861862962i\)
\(L(\frac12)\) \(\approx\) \(1.786872710 + 0.005861862962i\)
\(L(1)\) \(\approx\) \(1.037535814 - 0.4805350754i\)
\(L(1)\) \(\approx\) \(1.037535814 - 0.4805350754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.0950 - 0.995i)T \)
3 \( 1 + (0.981 + 0.189i)T \)
5 \( 1 + (-0.281 - 0.959i)T \)
7 \( 1 + (-0.971 - 0.235i)T \)
11 \( 1 + (0.971 - 0.235i)T \)
17 \( 1 + (0.995 + 0.0950i)T \)
19 \( 1 + (-0.371 + 0.928i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.723 + 0.690i)T \)
31 \( 1 + (-0.989 - 0.142i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.945 - 0.327i)T \)
43 \( 1 + (-0.723 + 0.690i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.189 - 0.981i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
67 \( 1 + (-0.189 + 0.981i)T \)
71 \( 1 + (0.690 + 0.723i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
97 \( 1 + (0.971 + 0.235i)T \)
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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

−21.36482729730331412066735394867, −19.8246169845075626716351723881, −19.44670302774242917210541526455, −18.80633258345244667164965503210, −18.13676028457963305923700311040, −17.168721131704771454606601009240, −16.29756884033643833417413644966, −15.255831104969341747027789610669, −15.154610500349423572228358482544, −14.1688367954907981846774639815, −13.61734538410137461341434864245, −12.73140346348358338286104309165, −11.85736874367278868673235017838, −10.44644629397028289427644645261, −9.6701639755583623901609250116, −9.09633960225507653737812888227, −8.22111754232581534073675577017, −7.17270133213204879550451850812, −6.87980674920470738498330130464, −6.05176010356593292195452077388, −4.74401531957591701630849339420, −3.50130164425367057963111610527, −3.29025188427928650525454846697, −1.81969726427103811462644224574, −0.36726616381445653410363335347, 1.01803457036268689213822703239, 1.667644458658333376146087511099, 3.08388804819053913842751239139, 3.58359065642871022146035912152, 4.34395517567285154919993361522, 5.31261954358303290329767215654, 6.669854822922649308902609785318, 7.87133686068146110571816097590, 8.58511563486121122750852073080, 9.226338380399297878882801365272, 9.86797306705231785131889628147, 10.62442480787279982774728967231, 11.85591661788347810891359646167, 12.61876522490684120483262092045, 12.98507547696690758718326582223, 14.03216402704575379995955789796, 14.53215803029528968031532383298, 15.7031558883543771132831603292, 16.73195976980080771526968192886, 16.9165614678121616996970649676, 18.502880598659782442212223139263, 19.05385097773985502252218983970, 19.70714989171890731094466248643, 20.20220849090243118669830286961, 20.91723514672073297707792700177

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