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Properties

Label 1-373-373.121-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.804 - 0.593i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (−0.409 + 0.912i)2-s + (−0.999 + 0.0337i)3-s + (−0.664 − 0.747i)4-s + (−0.780 − 0.625i)5-s + (0.378 − 0.925i)6-s + (0.347 − 0.937i)7-s + (0.954 − 0.299i)8-s + (0.997 − 0.0675i)9-s + (0.890 − 0.455i)10-s + (−0.217 − 0.975i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (0.713 + 0.701i)14-s + (0.801 + 0.598i)15-s + (−0.117 + 0.993i)16-s + (0.979 − 0.201i)17-s + ⋯
L(s)  = 1  + (−0.409 + 0.912i)2-s + (−0.999 + 0.0337i)3-s + (−0.664 − 0.747i)4-s + (−0.780 − 0.625i)5-s + (0.378 − 0.925i)6-s + (0.347 − 0.937i)7-s + (0.954 − 0.299i)8-s + (0.997 − 0.0675i)9-s + (0.890 − 0.455i)10-s + (−0.217 − 0.975i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (0.713 + 0.701i)14-s + (0.801 + 0.598i)15-s + (−0.117 + 0.993i)16-s + (0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.804 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06592582257 - 0.2003843538i\)
\(L(\frac12)\) \(\approx\) \(0.06592582257 - 0.2003843538i\)
\(L(1)\) \(\approx\) \(0.4478642162 + 0.009367228089i\)
\(L(1)\) \(\approx\) \(0.4478642162 + 0.009367228089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.409 + 0.912i)T \)
3 \( 1 + (-0.999 + 0.0337i)T \)
5 \( 1 + (-0.780 - 0.625i)T \)
7 \( 1 + (0.347 - 0.937i)T \)
11 \( 1 + (-0.217 - 0.975i)T \)
13 \( 1 + (-0.954 - 0.299i)T \)
17 \( 1 + (0.979 - 0.201i)T \)
19 \( 1 + (-0.918 - 0.394i)T \)
23 \( 1 + (-0.528 + 0.848i)T \)
29 \( 1 + (0.857 + 0.514i)T \)
31 \( 1 + (0.688 - 0.724i)T \)
37 \( 1 + (-0.905 - 0.425i)T \)
41 \( 1 + (-0.612 - 0.790i)T \)
43 \( 1 + (0.664 + 0.747i)T \)
47 \( 1 + (-0.963 - 0.266i)T \)
53 \( 1 + (-0.997 - 0.0675i)T \)
59 \( 1 + (0.0168 - 0.999i)T \)
61 \( 1 + (0.999 - 0.0337i)T \)
67 \( 1 + (-0.151 + 0.988i)T \)
71 \( 1 + (-0.315 - 0.948i)T \)
73 \( 1 + (-0.557 + 0.830i)T \)
79 \( 1 + (-0.890 + 0.455i)T \)
83 \( 1 + (0.997 + 0.0675i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.979 + 0.201i)T \)
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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

−25.0147684895563994608547607519, −23.75436134399307295588714365850, −22.97608968167710133177823959588, −22.30350911453130503462833524218, −21.52719658288948440900915712809, −20.72180738597036700219440138716, −19.33930478524164580161009328925, −18.8955470305083743336436242233, −18.01181845782523260029810602544, −17.33290159988846577400619480429, −16.28492992483383825226556168601, −15.223553057363953362915845029876, −14.313275034167453829165586969331, −12.55124281611801586470302185853, −12.20433417538449499550493194357, −11.580241421070253044324529510230, −10.392307350822435549386377883219, −9.96409092829987095625224602696, −8.43664790226711920254783042778, −7.56744217212634085076274770289, −6.4765224495393796828463315071, −4.967722121029785654701819377655, −4.27790427324980988958652672862, −2.79608049919236853894461389392, −1.72563858249513088633603639957, 0.19302347561520966497583175204, 1.20961279250704546621357297660, 3.802596770905038307332679931548, 4.79548505758146020410823525364, 5.47449259598083025690686477517, 6.7188665212711825194305664476, 7.622298852759608559350953179144, 8.299011504624634604123394095474, 9.699851895345876512933487872244, 10.561644339940934190392091588447, 11.465502381823102393659667474521, 12.5702005542062540393328557782, 13.56495681296364465134522505863, 14.650035419006073424853245634, 15.77703097209558341593763181139, 16.331704198890571309384053616242, 17.16690137729395385694902525035, 17.6166681754755885129749184095, 19.00078310881742031242167169742, 19.48578890436340406567561167988, 20.79027870910971827892729546072, 21.87861604207221885148143617078, 22.95089163436458957346917333022, 23.62491146153198177676005148623, 24.04976572119868932967622311534

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