Properties

Label 1-373-373.23-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.497 - 0.867i$
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  + (0.937 − 0.347i)2-s + (−0.918 − 0.394i)3-s + (0.758 − 0.651i)4-s + (−0.968 + 0.250i)5-s + (−0.998 − 0.0506i)6-s + (0.440 − 0.897i)7-s + (0.485 − 0.874i)8-s + (0.688 + 0.724i)9-s + (−0.820 + 0.571i)10-s + (0.485 + 0.874i)11-s + (−0.954 + 0.299i)12-s + (0.874 − 0.485i)13-s + (0.101 − 0.994i)14-s + (0.988 + 0.151i)15-s + (0.151 − 0.988i)16-s + (0.758 − 0.651i)17-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)2-s + (−0.918 − 0.394i)3-s + (0.758 − 0.651i)4-s + (−0.968 + 0.250i)5-s + (−0.998 − 0.0506i)6-s + (0.440 − 0.897i)7-s + (0.485 − 0.874i)8-s + (0.688 + 0.724i)9-s + (−0.820 + 0.571i)10-s + (0.485 + 0.874i)11-s + (−0.954 + 0.299i)12-s + (0.874 − 0.485i)13-s + (0.101 − 0.994i)14-s + (0.988 + 0.151i)15-s + (0.151 − 0.988i)16-s + (0.758 − 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.497 - 0.867i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.497 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.209136567 - 2.086643639i\)
\(L(\frac12)\) \(\approx\) \(1.209136567 - 2.086643639i\)
\(L(1)\) \(\approx\) \(1.229615465 - 0.6962578320i\)
\(L(1)\) \(\approx\) \(1.229615465 - 0.6962578320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.937 - 0.347i)T \)
3 \( 1 + (-0.918 - 0.394i)T \)
5 \( 1 + (-0.968 + 0.250i)T \)
7 \( 1 + (0.440 - 0.897i)T \)
11 \( 1 + (0.485 + 0.874i)T \)
13 \( 1 + (0.874 - 0.485i)T \)
17 \( 1 + (0.758 - 0.651i)T \)
19 \( 1 + (-0.988 + 0.151i)T \)
23 \( 1 + (0.394 + 0.918i)T \)
29 \( 1 + (0.979 - 0.201i)T \)
31 \( 1 + (0.954 + 0.299i)T \)
37 \( 1 + (-0.528 - 0.848i)T \)
41 \( 1 + (-0.0506 + 0.998i)T \)
43 \( 1 + (-0.651 - 0.758i)T \)
47 \( 1 + (-0.101 - 0.994i)T \)
53 \( 1 + (-0.724 - 0.688i)T \)
59 \( 1 + (-0.979 + 0.201i)T \)
61 \( 1 + (0.394 - 0.918i)T \)
67 \( 1 + (-0.968 + 0.250i)T \)
71 \( 1 + (0.758 + 0.651i)T \)
73 \( 1 + (0.688 - 0.724i)T \)
79 \( 1 + (-0.571 - 0.820i)T \)
83 \( 1 + (0.688 - 0.724i)T \)
89 \( 1 - T \)
97 \( 1 + (0.651 + 0.758i)T \)
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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

−24.22236373523341501561916103074, −23.82016718232758666118815920838, −22.9782979475817193347783781349, −22.20375976571845887810573144021, −21.233791356945159359014915032677, −20.92391674289863858470283826147, −19.37643053685013061044302468052, −18.57489212464062237536396748615, −17.17365252769058096332141690116, −16.54699785567945296519427932317, −15.701010161958319136032113513566, −15.11740919140247334318069275838, −14.11647875203922362494751677345, −12.70375820222034131389426413168, −12.090386785153783588521223404999, −11.35477247563920197963370367724, −10.681473347739930030591045752173, −8.777852118907862778830047680204, −8.1498760490265863051699192834, −6.59077326654284132540145100134, −6.03291161394921715668697759404, −4.868132017773953019003562843559, −4.157231978934825960345965766060, −3.11158418954185160752169121903, −1.249671747833233332737252068435, 0.63344358560365568575998253263, 1.64288776596678135507192547640, 3.36200243079582126050196758738, 4.32669273799150849383004214667, 5.0774812694624387938380807011, 6.43803957966909342983066846287, 7.127252049731025569587838762349, 8.01741176901138415185768840857, 10.08048745850953508498442627029, 10.7838528919559308574760925009, 11.60124620777556482882295683917, 12.22601332695943763490544621292, 13.19423718108887969726509575004, 14.129051544690738418075580755439, 15.17139720497877960335004508381, 15.97144024099796223522982302695, 16.9334602063477594274201425592, 17.943830978515808641596792963278, 19.03651776208219661375181216007, 19.77438923317638849756413315915, 20.685322044146213640688636324117, 21.581830959425595106795903353158, 22.86839777772380251282180254496, 23.17995897578673733069071459661, 23.51711270648340255473302078718

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