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Properties

Label 1-373-373.142-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.343 - 0.939i$
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  + (−0.485 + 0.874i)2-s + (−0.347 − 0.937i)3-s + (−0.528 − 0.848i)4-s + (0.724 − 0.688i)5-s + (0.988 + 0.151i)6-s + (−0.979 + 0.201i)7-s + (0.998 − 0.0506i)8-s + (−0.758 + 0.651i)9-s + (0.250 + 0.968i)10-s + (0.998 + 0.0506i)11-s + (−0.612 + 0.790i)12-s + (0.0506 − 0.998i)13-s + (0.299 − 0.954i)14-s + (−0.897 − 0.440i)15-s + (−0.440 + 0.897i)16-s + (−0.528 − 0.848i)17-s + ⋯
L(s)  = 1  + (−0.485 + 0.874i)2-s + (−0.347 − 0.937i)3-s + (−0.528 − 0.848i)4-s + (0.724 − 0.688i)5-s + (0.988 + 0.151i)6-s + (−0.979 + 0.201i)7-s + (0.998 − 0.0506i)8-s + (−0.758 + 0.651i)9-s + (0.250 + 0.968i)10-s + (0.998 + 0.0506i)11-s + (−0.612 + 0.790i)12-s + (0.0506 − 0.998i)13-s + (0.299 − 0.954i)14-s + (−0.897 − 0.440i)15-s + (−0.440 + 0.897i)16-s + (−0.528 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.343 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6471894460 - 0.9253523581i\)
\(L(\frac12)\) \(\approx\) \(0.6471894460 - 0.9253523581i\)
\(L(1)\) \(\approx\) \(0.7645407301 - 0.1835828144i\)
\(L(1)\) \(\approx\) \(0.7645407301 - 0.1835828144i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.485 + 0.874i)T \)
3 \( 1 + (-0.347 - 0.937i)T \)
5 \( 1 + (0.724 - 0.688i)T \)
7 \( 1 + (-0.979 + 0.201i)T \)
11 \( 1 + (0.998 + 0.0506i)T \)
13 \( 1 + (0.0506 - 0.998i)T \)
17 \( 1 + (-0.528 - 0.848i)T \)
19 \( 1 + (0.897 - 0.440i)T \)
23 \( 1 + (0.937 + 0.347i)T \)
29 \( 1 + (0.820 - 0.571i)T \)
31 \( 1 + (0.612 + 0.790i)T \)
37 \( 1 + (0.994 - 0.101i)T \)
41 \( 1 + (0.151 - 0.988i)T \)
43 \( 1 + (-0.848 + 0.528i)T \)
47 \( 1 + (-0.299 - 0.954i)T \)
53 \( 1 + (-0.651 + 0.758i)T \)
59 \( 1 + (-0.820 + 0.571i)T \)
61 \( 1 + (0.937 - 0.347i)T \)
67 \( 1 + (0.724 - 0.688i)T \)
71 \( 1 + (-0.528 + 0.848i)T \)
73 \( 1 + (-0.758 - 0.651i)T \)
79 \( 1 + (-0.968 + 0.250i)T \)
83 \( 1 + (-0.758 - 0.651i)T \)
89 \( 1 - T \)
97 \( 1 + (0.848 - 0.528i)T \)
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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

−25.0688725759793739822277596822, −23.31710491996932520971217669629, −22.44642968149666472399663408593, −21.994591094341383271146818250881, −21.31516247244328273000442607480, −20.30630675616182182215725672395, −19.43011088176043860327602887390, −18.62342656179703747496728137319, −17.501441580404306771535129725170, −16.87087908028452779059126710291, −16.13048350306834277531734872595, −14.73245880245553840365674110557, −13.89782148921359927564622445172, −12.84562517629238061684534879819, −11.65587505594519394763617882606, −10.998831266514749406179482931061, −9.9227090685778258124474616029, −9.57690049249241884720827103938, −8.651842420846839483213581635439, −6.871379156127572767539497004679, −6.13985537596638987923620123038, −4.557156093694253771576930768191, −3.610267481206951096615780257178, −2.76202843906089619384301543839, −1.2476830839643062829320987624, 0.47065335797009572081123282915, 1.26195088636596549841766423328, 2.8104066530057316179829088842, 4.82013021832495202022822012707, 5.70577989409621815343664860786, 6.49908000139572243130784583587, 7.251564447581972774599385981997, 8.52055623191340477763371208370, 9.26557689114861700329585968822, 10.134530219831871245705541765458, 11.54686702718518811043597713736, 12.691785234284029497630982153253, 13.43677466911807024545899353476, 14.083585720887721381568245981791, 15.4941093655065784794519280323, 16.32489209370367468730465457391, 17.20416020142221762874978176753, 17.74273691194533788664058603276, 18.59166555966095951194955067701, 19.68041203386038542675014859003, 20.10447277792737390981612653951, 21.89804010002483712583771949772, 22.68406101353717691213637584542, 23.30704778290156165093891340520, 24.601727957363372151507579431137

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