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Properties

Label 1-1375-1375.1142-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.429 - 0.902i$
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  + (−0.481 + 0.876i)2-s + (0.844 − 0.535i)3-s + (−0.535 − 0.844i)4-s + (0.0627 + 0.998i)6-s + i·7-s + (0.998 − 0.0627i)8-s + (0.425 − 0.904i)9-s + (−0.904 − 0.425i)12-s + (0.904 + 0.425i)13-s + (−0.876 − 0.481i)14-s + (−0.425 + 0.904i)16-s + (−0.368 − 0.929i)17-s + (0.587 + 0.809i)18-s + (0.929 − 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (0.844 − 0.535i)3-s + (−0.535 − 0.844i)4-s + (0.0627 + 0.998i)6-s + i·7-s + (0.998 − 0.0627i)8-s + (0.425 − 0.904i)9-s + (−0.904 − 0.425i)12-s + (0.904 + 0.425i)13-s + (−0.876 − 0.481i)14-s + (−0.425 + 0.904i)16-s + (−0.368 − 0.929i)17-s + (0.587 + 0.809i)18-s + (0.929 − 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.429 - 0.902i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.429 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4068739037 - 0.6443347681i\)
\(L(\frac12)\) \(\approx\) \(0.4068739037 - 0.6443347681i\)
\(L(1)\) \(\approx\) \(0.9490316254 + 0.1323651255i\)
\(L(1)\) \(\approx\) \(0.9490316254 + 0.1323651255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (0.844 - 0.535i)T \)
7 \( 1 + iT \)
13 \( 1 + (0.904 + 0.425i)T \)
17 \( 1 + (-0.368 - 0.929i)T \)
19 \( 1 + (0.929 - 0.368i)T \)
23 \( 1 + (-0.684 - 0.728i)T \)
29 \( 1 + (-0.968 + 0.248i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (-0.684 + 0.728i)T \)
41 \( 1 + (0.876 - 0.481i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.770 - 0.637i)T \)
53 \( 1 + (-0.998 - 0.0627i)T \)
59 \( 1 + (0.992 + 0.125i)T \)
61 \( 1 + (-0.187 + 0.982i)T \)
67 \( 1 + (-0.998 + 0.0627i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (-0.684 - 0.728i)T \)
79 \( 1 + (-0.968 + 0.248i)T \)
83 \( 1 + (0.248 - 0.968i)T \)
89 \( 1 + (0.425 + 0.904i)T \)
97 \( 1 + (-0.770 - 0.637i)T \)
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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

−20.69338688748148935255938478189, −20.14831965536298778746890989868, −19.6338213955054827417330162886, −18.84684727671757073673710508422, −17.949813329241281937374356319890, −17.26927344773844115265496585736, −16.28307811762513526280126937888, −15.81772139465500122269668444477, −14.588372858050506422187797131769, −13.868711163870904887561541407261, −13.26066882891211661372858887648, −12.57139635342975692608055754415, −11.24886403709876176543271974436, −10.793708089415537494815847570185, −10.00129177874683988793059814812, −9.40258594795862900040044564746, −8.44784891703302879729094316661, −7.865607654720344102292353515487, −7.10594731365379108333824461689, −5.608489885337977745829810829653, −4.45815488812663834519815417742, −3.62149156149180742174147292645, −3.35107886628670693938603489443, −1.93720794067870617855167977773, −1.27738785064986605409143215112, 0.14937219353641427265051456436, 1.38398267347851213346473366689, 2.229791868390652566003192661303, 3.33315198281687234101077573788, 4.4703839391161844373606759837, 5.52897148110031140896788008700, 6.32246232921004481355756506828, 7.079676378167233271777275173114, 7.89546476181379788921044957136, 8.67493931919961605372215491086, 9.20341761829970139245640291286, 9.83546218811927335709538960653, 11.16138737667273828503079386454, 11.95751590785702747904440577328, 13.09710388683090981449057695602, 13.62188340539888062778261471274, 14.4716776012109138551707861550, 15.04969275217643997829323403718, 15.9233264711540149823024823712, 16.324137290140310249603502481, 17.6727995355058123143423449590, 18.32563573927828366060270924110, 18.58397025875012969508595362941, 19.43708710297460897167717064802, 20.29130478913385852023671541247

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