Properties

Label 1-2368-2368.1179-r0-0-0
Degree $1$
Conductor $2368$
Sign $0.542 + 0.840i$
Analytic cond. $10.9969$
Root an. cond. $10.9969$
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  + (0.300 − 0.953i)3-s + (0.0436 − 0.999i)5-s + (−0.906 − 0.422i)7-s + (−0.819 − 0.573i)9-s + (−0.991 + 0.130i)11-s + (0.976 − 0.216i)13-s + (−0.939 − 0.342i)15-s + (−0.173 − 0.984i)17-s + (−0.300 + 0.953i)19-s + (−0.675 + 0.737i)21-s + (−0.965 + 0.258i)23-s + (−0.996 − 0.0871i)25-s + (−0.793 + 0.608i)27-s + (0.793 + 0.608i)29-s + i·31-s + ⋯
L(s)  = 1  + (0.300 − 0.953i)3-s + (0.0436 − 0.999i)5-s + (−0.906 − 0.422i)7-s + (−0.819 − 0.573i)9-s + (−0.991 + 0.130i)11-s + (0.976 − 0.216i)13-s + (−0.939 − 0.342i)15-s + (−0.173 − 0.984i)17-s + (−0.300 + 0.953i)19-s + (−0.675 + 0.737i)21-s + (−0.965 + 0.258i)23-s + (−0.996 − 0.0871i)25-s + (−0.793 + 0.608i)27-s + (0.793 + 0.608i)29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(10.9969\)
Root analytic conductor: \(10.9969\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2368,\ (0:\ ),\ 0.542 + 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1574770753 + 0.08581302891i\)
\(L(\frac12)\) \(\approx\) \(0.1574770753 + 0.08581302891i\)
\(L(1)\) \(\approx\) \(0.6981766183 - 0.4007456128i\)
\(L(1)\) \(\approx\) \(0.6981766183 - 0.4007456128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.300 - 0.953i)T \)
5 \( 1 + (0.0436 - 0.999i)T \)
7 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (-0.991 + 0.130i)T \)
13 \( 1 + (0.976 - 0.216i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.300 + 0.953i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (0.793 + 0.608i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.573 + 0.819i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.675 + 0.737i)T \)
59 \( 1 + (-0.675 + 0.737i)T \)
61 \( 1 + (-0.216 - 0.976i)T \)
67 \( 1 + (0.675 + 0.737i)T \)
71 \( 1 + (0.0871 + 0.996i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.537 - 0.843i)T \)
89 \( 1 + (0.906 - 0.422i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−19.434817926545969387830397006751, −18.84964994477447518106868699480, −18.10859788547212727027845450868, −17.328667209204119901969953338810, −16.315414260930539150857126055661, −15.72425794095474810129014874355, −15.3379248596982251449882045134, −14.60525218054943394625550367038, −13.63083115005355999081310145057, −13.26283924910384240441942447023, −12.15082321441731642704347852007, −11.07753155649521712338142701293, −10.76938886917501522127563008567, −9.93987014423105271390346722008, −9.40382580926913204629195847035, −8.3751048326859718305709100693, −7.89720843650592453590871975768, −6.484738942129137028532996634993, −6.1972714775457702690303594225, −5.240179208935472598435569321005, −4.12639156231509029397567848977, −3.52142209180953139869815660786, −2.70087276732280024913579823977, −2.13643216534221262862512707815, −0.05937939404688757478226368812, 1.01065447373642486082238687220, 1.82001001130081146670991296168, 2.92540121246291963567373432627, 3.59742166983625007414592383834, 4.70253365055119562522324563502, 5.63941932922010125217411832201, 6.29676454062430405524979751361, 7.12649469297041279061961899024, 8.031688169264609194557727884956, 8.43328424347188346617709413763, 9.36580316919503680081114073089, 10.07664131334552098058486920255, 10.99247786605628052174329194099, 12.104956243049391727615182900, 12.47907516528440099593013324749, 13.31961751679213071286068402366, 13.58172529867932851578696912653, 14.40916405456026927953924404974, 15.69657875326832958366986861271, 16.06810946985725560893873800920, 16.76373690490482500547371679050, 17.74054563570331016603883927652, 18.26738826952304093994154264641, 18.91361550033364896941057246352, 19.95543830576613886884426006252

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