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Properties

Label 1-1375-1375.922-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.918 + 0.395i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.904 − 0.425i)2-s + (−0.844 − 0.535i)3-s + (0.637 + 0.770i)4-s + (0.535 + 0.844i)6-s + (−0.587 − 0.809i)7-s + (−0.248 − 0.968i)8-s + (0.425 + 0.904i)9-s + (−0.125 − 0.992i)12-s + (−0.904 + 0.425i)13-s + (0.187 + 0.982i)14-s + (−0.187 + 0.982i)16-s + (0.248 + 0.968i)17-s − i·18-s + (−0.0627 + 0.998i)19-s + (0.0627 + 0.998i)21-s + ⋯
L(s)  = 1  + (−0.904 − 0.425i)2-s + (−0.844 − 0.535i)3-s + (0.637 + 0.770i)4-s + (0.535 + 0.844i)6-s + (−0.587 − 0.809i)7-s + (−0.248 − 0.968i)8-s + (0.425 + 0.904i)9-s + (−0.125 − 0.992i)12-s + (−0.904 + 0.425i)13-s + (0.187 + 0.982i)14-s + (−0.187 + 0.982i)16-s + (0.248 + 0.968i)17-s − i·18-s + (−0.0627 + 0.998i)19-s + (0.0627 + 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.918 + 0.395i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (922, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03955753292 + 0.1918746247i\)
\(L(\frac12)\) \(\approx\) \(0.03955753292 + 0.1918746247i\)
\(L(1)\) \(\approx\) \(0.4498740610 - 0.06286104005i\)
\(L(1)\) \(\approx\) \(0.4498740610 - 0.06286104005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.904 - 0.425i)T \)
3 \( 1 + (-0.844 - 0.535i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.904 + 0.425i)T \)
17 \( 1 + (0.248 + 0.968i)T \)
19 \( 1 + (-0.0627 + 0.998i)T \)
23 \( 1 + (0.904 + 0.425i)T \)
29 \( 1 + (-0.535 + 0.844i)T \)
31 \( 1 + (0.535 + 0.844i)T \)
37 \( 1 + (-0.481 + 0.876i)T \)
41 \( 1 + (-0.992 + 0.125i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.770 - 0.637i)T \)
53 \( 1 + (-0.844 - 0.535i)T \)
59 \( 1 + (0.425 + 0.904i)T \)
61 \( 1 + (-0.187 - 0.982i)T \)
67 \( 1 + (-0.770 - 0.637i)T \)
71 \( 1 + (0.0627 + 0.998i)T \)
73 \( 1 + (-0.982 + 0.187i)T \)
79 \( 1 + (0.637 + 0.770i)T \)
83 \( 1 + (0.770 + 0.637i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
97 \( 1 + (-0.998 + 0.0627i)T \)
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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

−20.38737722975408421136253957875, −19.170610547101359087583895131552, −18.8591211114813049882333166595, −17.824097464836596699106943041060, −17.32891522526498320026126299391, −16.59662050422984804298077522387, −15.8303463769391742603298687195, −15.3038905008711070907272901562, −14.70318765876549980799722749799, −13.39224167460454700022200109784, −12.28269221207013653745294521797, −11.737960958946553778558313941273, −10.87864570490752630952479645721, −10.108381062159297776349425632455, −9.332173295188110778546430268536, −8.96191740520088696312492489737, −7.595248467341199630598575707842, −6.91586699402135440596111364435, −6.04501467974765559008358599743, −5.3507552288671564884631932796, −4.63110097523957417586928523104, −3.07586272912029697961790180437, −2.26480773457990083088835443107, −0.67481317504212857204460828968, −0.09507657966640508303356908071, 1.117970181906269897649302913089, 1.74153937561221570762230582833, 3.02432693190513572684796159889, 3.958667988311921641672864790122, 5.08903266004512937200910555978, 6.27901429439075314297615702739, 6.91341892335208300276022571510, 7.552198788558555425429410282039, 8.40078860168948996268325376198, 9.54443850918006044869059063769, 10.27704669751079697525952615174, 10.75791922812469440002144484990, 11.71257529070121976381871074501, 12.43034906026067015119346639973, 12.94219120363103537988881449339, 13.90056177499540758064579773077, 15.05915762859195730717497576084, 16.136385949830417052839767642122, 16.80009671144282835995702723829, 17.12028705679571148322537276912, 17.89403168378587036306137555007, 18.93741395087320112374492636752, 19.19153162716401546785275965015, 19.97352859684205666693617627990, 20.8939676873613443296599522770

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