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Properties

Label 1-373-373.62-r1-0-0
Degree $1$
Conductor $373$
Sign $0.422 - 0.906i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.625 − 0.780i)2-s + (0.839 + 0.543i)3-s + (−0.217 − 0.975i)4-s + (0.882 − 0.470i)5-s + (0.948 − 0.315i)6-s + (0.250 − 0.968i)7-s + (−0.897 − 0.440i)8-s + (0.409 + 0.912i)9-s + (0.184 − 0.982i)10-s + (0.830 + 0.557i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (−0.598 − 0.801i)14-s + (0.996 + 0.0843i)15-s + (−0.905 + 0.425i)16-s + (0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (0.625 − 0.780i)2-s + (0.839 + 0.543i)3-s + (−0.217 − 0.975i)4-s + (0.882 − 0.470i)5-s + (0.948 − 0.315i)6-s + (0.250 − 0.968i)7-s + (−0.897 − 0.440i)8-s + (0.409 + 0.912i)9-s + (0.184 − 0.982i)10-s + (0.830 + 0.557i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (−0.598 − 0.801i)14-s + (0.996 + 0.0843i)15-s + (−0.905 + 0.425i)16-s + (0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.422 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.248875799 - 2.708653244i\)
\(L(\frac12)\) \(\approx\) \(4.248875799 - 2.708653244i\)
\(L(1)\) \(\approx\) \(2.246278080 - 0.9640976520i\)
\(L(1)\) \(\approx\) \(2.246278080 - 0.9640976520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.625 - 0.780i)T \)
3 \( 1 + (0.839 + 0.543i)T \)
5 \( 1 + (0.882 - 0.470i)T \)
7 \( 1 + (0.250 - 0.968i)T \)
11 \( 1 + (0.830 + 0.557i)T \)
13 \( 1 + (0.440 + 0.897i)T \)
17 \( 1 + (0.954 + 0.299i)T \)
19 \( 1 + (0.571 + 0.820i)T \)
23 \( 1 + (0.998 - 0.0506i)T \)
29 \( 1 + (-0.972 - 0.234i)T \)
31 \( 1 + (-0.347 - 0.937i)T \)
37 \( 1 + (0.378 - 0.925i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (-0.975 + 0.217i)T \)
47 \( 1 + (-0.993 - 0.117i)T \)
53 \( 1 + (-0.912 - 0.409i)T \)
59 \( 1 + (-0.283 - 0.959i)T \)
61 \( 1 + (-0.543 + 0.839i)T \)
67 \( 1 + (-0.848 - 0.528i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.585 + 0.810i)T \)
79 \( 1 + (0.982 + 0.184i)T \)
83 \( 1 + (0.409 - 0.912i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.299 + 0.954i)T \)
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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

−24.903025208812172770827801593630, −23.93861642666826410462139734542, −22.790478704791596870821777087686, −21.957514677712818826058434750940, −21.24795015677719299045837898102, −20.42957928001836281862907046234, −19.060149709018614637698365564823, −18.22651027727442781507834311180, −17.64732401253093850910611545378, −16.47459681845255725254864628240, −15.229163858729938053547127429149, −14.73197547027451606599312062143, −13.89794941434426706290569036059, −13.181394602088562605623414395776, −12.29792174442890351888129845847, −11.208168742267560591360077003801, −9.45552484435088850001163702422, −8.87749406659583116999524803409, −7.85095133091315360415199834156, −6.833786788170206709564243908136, −5.97197849295840227859231817458, −5.12334437820572701732685011204, −3.29246494150816959392800733568, −2.88725356439570690610292811316, −1.371474314497773135697850290641, 1.27920586214698732683182637360, 1.91377718817064685752018995415, 3.4368577362651086058509954806, 4.1820002418245293316611605448, 5.0850080826482837566196397920, 6.32104035890247875034566254910, 7.69572566759949657311907732049, 9.18642368312758289057866351550, 9.60098542764404016315616741510, 10.50117696280128599480639533028, 11.51255640806829121010124558848, 12.81547697731370938382535409904, 13.50468887807482970666851445759, 14.42301028309076757665915250303, 14.68032976974006152241929662174, 16.3153849654085780998458036431, 16.98101153158629038598926596550, 18.33271427539690913369472913848, 19.3161024225292278624790612239, 20.16380596781192616158692759861, 20.89995769303759904736695589090, 21.214366432042835391516052809691, 22.321575474193084669144481915892, 23.16889386895026648228933276224, 24.28307901324432423524805840885

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