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Properties

Label 1-373-373.57-r1-0-0
Degree $1$
Conductor $373$
Sign $0.314 + 0.949i$
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  + (−0.234 − 0.972i)2-s + (−0.963 − 0.266i)3-s + (−0.890 + 0.455i)4-s + (−0.769 + 0.638i)5-s + (−0.0337 + 0.999i)6-s + (0.954 + 0.299i)7-s + (0.651 + 0.758i)8-s + (0.857 + 0.514i)9-s + (0.801 + 0.598i)10-s + (−0.982 − 0.184i)11-s + (0.979 − 0.201i)12-s + (0.758 + 0.651i)13-s + (0.0675 − 0.997i)14-s + (0.912 − 0.409i)15-s + (0.585 − 0.810i)16-s + (0.0506 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.234 − 0.972i)2-s + (−0.963 − 0.266i)3-s + (−0.890 + 0.455i)4-s + (−0.769 + 0.638i)5-s + (−0.0337 + 0.999i)6-s + (0.954 + 0.299i)7-s + (0.651 + 0.758i)8-s + (0.857 + 0.514i)9-s + (0.801 + 0.598i)10-s + (−0.982 − 0.184i)11-s + (0.979 − 0.201i)12-s + (0.758 + 0.651i)13-s + (0.0675 − 0.997i)14-s + (0.912 − 0.409i)15-s + (0.585 − 0.810i)16-s + (0.0506 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4708502550 + 0.3400327909i\)
\(L(\frac12)\) \(\approx\) \(0.4708502550 + 0.3400327909i\)
\(L(1)\) \(\approx\) \(0.5828811605 - 0.1270927467i\)
\(L(1)\) \(\approx\) \(0.5828811605 - 0.1270927467i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.234 - 0.972i)T \)
3 \( 1 + (-0.963 - 0.266i)T \)
5 \( 1 + (-0.769 + 0.638i)T \)
7 \( 1 + (0.954 + 0.299i)T \)
11 \( 1 + (-0.982 - 0.184i)T \)
13 \( 1 + (0.758 + 0.651i)T \)
17 \( 1 + (0.0506 - 0.998i)T \)
19 \( 1 + (0.101 + 0.994i)T \)
23 \( 1 + (0.968 + 0.250i)T \)
29 \( 1 + (-0.378 + 0.925i)T \)
31 \( 1 + (-0.979 - 0.201i)T \)
37 \( 1 + (0.931 - 0.363i)T \)
41 \( 1 + (0.528 - 0.848i)T \)
43 \( 1 + (-0.455 - 0.890i)T \)
47 \( 1 + (-0.830 + 0.557i)T \)
53 \( 1 + (0.514 + 0.857i)T \)
59 \( 1 + (-0.990 + 0.134i)T \)
61 \( 1 + (-0.266 + 0.963i)T \)
67 \( 1 + (0.937 + 0.347i)T \)
71 \( 1 + (0.839 - 0.543i)T \)
73 \( 1 + (0.0168 + 0.999i)T \)
79 \( 1 + (0.598 - 0.801i)T \)
83 \( 1 + (0.857 - 0.514i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.998 - 0.0506i)T \)
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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

−24.09742708277015093816871047950, −23.340974914784636498629910854999, −23.07935531729014977292340702104, −21.69255989719018866901981208598, −20.86336797058195300745860732203, −19.76956425978619310055355718044, −18.484533159989288690597338735817, −17.87412553720145442949863747368, −17.00784322721271813158178926491, −16.33735340767962066986751310182, −15.317774729618129317234685424877, −15.014586844930307775481195694992, −13.25473872708565036098071681678, −12.77192169809936726490694174487, −11.24926331363930124452394239960, −10.74848117163229937500763215114, −9.48238462758089276163388403255, −8.2453849286308715050131332976, −7.72171854100812978689097512830, −6.52401228636343547868421231296, −5.30148310757151518704715102894, −4.81851117819978684471442169839, −3.81231373804560389481329933339, −1.244135684208326117351880677314, −0.27119401952811917181039490283, 1.0763059162587821437513268491, 2.28724080574336327358040623271, 3.62531180611172433162244421643, 4.739036550792498888523115471956, 5.6293465415888445749033992581, 7.25109015698356600631737323535, 7.91059869205905023202917622945, 9.11144749019053409533162043268, 10.562760359072029645513346790850, 11.044489711444573939611235025681, 11.68264391071348911437240140588, 12.51525679065636666737333871501, 13.595214108750871634526711839612, 14.61879645346127399008805604314, 15.90471306050223136535214831367, 16.72742355689098463299958907774, 17.99596960622510342988724992425, 18.498433247330606310358358137859, 18.8729763652275313116638193761, 20.29424445699516375878799617210, 21.14167449488128062910797917853, 21.846140255258788200598447208105, 22.92653515648419684505099091836, 23.35660872420427911082300336177, 24.2087644463766734163471150799

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