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Properties

Label 1-373-373.42-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.751 + 0.659i$
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  + (0.455 + 0.890i)2-s + (−0.857 − 0.514i)3-s + (−0.585 + 0.810i)4-s + (−0.982 − 0.184i)5-s + (0.0675 − 0.997i)6-s + (−0.820 − 0.571i)7-s + (−0.988 − 0.151i)8-s + (0.470 + 0.882i)9-s + (−0.283 − 0.959i)10-s + (0.363 − 0.931i)11-s + (0.918 − 0.394i)12-s + (−0.151 − 0.988i)13-s + (0.134 − 0.990i)14-s + (0.747 + 0.664i)15-s + (−0.315 − 0.948i)16-s + (0.994 + 0.101i)17-s + ⋯
L(s)  = 1  + (0.455 + 0.890i)2-s + (−0.857 − 0.514i)3-s + (−0.585 + 0.810i)4-s + (−0.982 − 0.184i)5-s + (0.0675 − 0.997i)6-s + (−0.820 − 0.571i)7-s + (−0.988 − 0.151i)8-s + (0.470 + 0.882i)9-s + (−0.283 − 0.959i)10-s + (0.363 − 0.931i)11-s + (0.918 − 0.394i)12-s + (−0.151 − 0.988i)13-s + (0.134 − 0.990i)14-s + (0.747 + 0.664i)15-s + (−0.315 − 0.948i)16-s + (0.994 + 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06492809250 + 0.1723064335i\)
\(L(\frac12)\) \(\approx\) \(0.06492809250 + 0.1723064335i\)
\(L(1)\) \(\approx\) \(0.6087187881 + 0.08371216104i\)
\(L(1)\) \(\approx\) \(0.6087187881 + 0.08371216104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.455 + 0.890i)T \)
3 \( 1 + (-0.857 - 0.514i)T \)
5 \( 1 + (-0.982 - 0.184i)T \)
7 \( 1 + (-0.820 - 0.571i)T \)
11 \( 1 + (0.363 - 0.931i)T \)
13 \( 1 + (-0.151 - 0.988i)T \)
17 \( 1 + (0.994 + 0.101i)T \)
19 \( 1 + (-0.201 - 0.979i)T \)
23 \( 1 + (-0.485 + 0.874i)T \)
29 \( 1 + (-0.713 - 0.701i)T \)
31 \( 1 + (-0.918 - 0.394i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.440 - 0.897i)T \)
43 \( 1 + (-0.810 - 0.585i)T \)
47 \( 1 + (0.925 + 0.378i)T \)
53 \( 1 + (0.882 + 0.470i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (-0.514 + 0.857i)T \)
67 \( 1 + (0.651 - 0.758i)T \)
71 \( 1 + (-0.409 + 0.912i)T \)
73 \( 1 + (-0.999 + 0.0337i)T \)
79 \( 1 + (-0.959 + 0.283i)T \)
83 \( 1 + (0.470 - 0.882i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.101 - 0.994i)T \)
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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

−23.44031242888880864165782171205, −23.11885130429223008922581300941, −22.30171293861567480043668621033, −21.65423180114745443042127609729, −20.604285842523325733702753844048, −19.79040413563228411623532409760, −18.72540659640362981323875992808, −18.33850364073529617478467656349, −16.76151700973808658995875571409, −16.09114420117267062478797938357, −14.97230013966224224990583106709, −14.449403106574479311834471778124, −12.65662059553629934454236309745, −12.23355087064690498258769613082, −11.61095880518165097763430423111, −10.49776725220550259479000314074, −9.75788399926019477447553839457, −8.84001570989035615458139974794, −7.08808336913677322186456325318, −6.11971128417311265892394029981, −5.007365437869539518290385731845, −4.03353888762980683833169992697, −3.33662909834311592510205017008, −1.722926153010942083811331736945, −0.07823006823352790488456728588, 0.72912533913239592398153040068, 3.20233979727502443658670653172, 4.021348819715068896478646417728, 5.31843262393582347794197259799, 6.057962012700016638535013021194, 7.21965241594646345225387365764, 7.67657101412725762759140713132, 8.85363383844451133285670580120, 10.29880133942443825162688899003, 11.477090786108803564530656841998, 12.26594616088845365366112884200, 13.121039275172601836934343401701, 13.796957770782264818489987479007, 15.239260308727165394396560945076, 15.90177910348524958884762232052, 16.80562487698241919572389506858, 17.206150852980576075911880189354, 18.546239861665577841910383960979, 19.24231048820880362762826879643, 20.27209538999605858150729593327, 21.75806446174012913423639902121, 22.43110192731927015400968356259, 23.154809905516518411727835131327, 23.80918475965339666883285455525, 24.39509431221687341104195016304

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