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Properties

Label 1-388-388.343-r0-0-0
Degree $1$
Conductor $388$
Sign $0.622 + 0.782i$
Analytic cond. $1.80186$
Root an. cond. $1.80186$
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  + (−0.382 + 0.923i)3-s + (0.980 − 0.195i)5-s + (−0.980 − 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.195 + 0.980i)13-s + (−0.195 + 0.980i)15-s + (0.195 + 0.980i)17-s + (0.980 − 0.195i)19-s + (0.555 − 0.831i)21-s + (0.831 + 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (−0.831 − 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (0.980 − 0.195i)5-s + (−0.980 − 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.195 + 0.980i)13-s + (−0.195 + 0.980i)15-s + (0.195 + 0.980i)17-s + (0.980 − 0.195i)19-s + (0.555 − 0.831i)21-s + (0.831 + 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (−0.831 − 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $0.622 + 0.782i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ 0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126205929 + 0.5428925082i\)
\(L(\frac12)\) \(\approx\) \(1.126205929 + 0.5428925082i\)
\(L(1)\) \(\approx\) \(1.018130164 + 0.2737436623i\)
\(L(1)\) \(\approx\) \(1.018130164 + 0.2737436623i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.980 - 0.195i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.195 + 0.980i)T \)
17 \( 1 + (0.195 + 0.980i)T \)
19 \( 1 + (0.980 - 0.195i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (-0.831 - 0.555i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (0.555 + 0.831i)T \)
41 \( 1 + (-0.555 + 0.831i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.831 + 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.195 - 0.980i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.980 - 0.195i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
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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

−24.693141042116448482906555733874, −23.29401716925386103734660016579, −22.45152473272045473334126890766, −22.3095287922049882786748009535, −20.666120210772091918137688963371, −20.04912554927534093933098884758, −18.8075560366488916948118774822, −18.2881217645213915470623218215, −17.415748603756462002960321396520, −16.68824492429243408746220362993, −15.548590767934353024307549355536, −14.35549011477986139144328073065, −13.4854711451044223849326370862, −12.76601080774018134919343246052, −12.04633795921607648557054051714, −10.783488654682534873840600734024, −9.81859031234725237663815875256, −9.009527531395440058672295956782, −7.5040050245913538916825386243, −6.808538637073175991389649638128, −5.86649718608098725542067433250, −5.09619335344253829489960872305, −3.17159976757825971299561915180, −2.29591583150108432897001977960, −0.95727021967038322577683384210, 1.20433177981546565095376708314, 2.9610307607025786508481230581, 3.83618652861589572712337376700, 5.09407113937548442360503691604, 6.06686441323958103229428237444, 6.65809641383736197654494044419, 8.5090396287786096323802827763, 9.442584228759126808351266227977, 9.900232351477456350981492589354, 11.006910059380322573515956011935, 11.87294585185958863739594983487, 13.193929652096686316565282718297, 13.84361579817898939320816986009, 14.912103694903279757573148154029, 15.99540700881627562303318209772, 16.771212224902641959139391697763, 17.15154977164880277529437639862, 18.464007923042282058581871730038, 19.41512751858380736448124645306, 20.415878702129176249820191069057, 21.47047931190147386632403546696, 21.76121512166719402066312374056, 22.6891838271818195115681355782, 23.62137926710351089493754612144, 24.65301524184210772617660830747

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