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Properties

Label 1-373-373.51-r0-0-0
Degree $1$
Conductor $373$
Sign $0.415 - 0.909i$
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  + (0.990 − 0.134i)2-s + (0.736 − 0.676i)3-s + (0.963 − 0.266i)4-s + (−0.664 + 0.747i)5-s + (0.638 − 0.769i)6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (0.0843 − 0.996i)9-s + (−0.557 + 0.830i)10-s + (−0.117 − 0.993i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.184 − 0.982i)14-s + (0.0168 + 0.999i)15-s + (0.857 − 0.514i)16-s + (−0.250 + 0.968i)17-s + ⋯
L(s)  = 1  + (0.990 − 0.134i)2-s + (0.736 − 0.676i)3-s + (0.963 − 0.266i)4-s + (−0.664 + 0.747i)5-s + (0.638 − 0.769i)6-s + (−0.0506 − 0.998i)7-s + (0.918 − 0.394i)8-s + (0.0843 − 0.996i)9-s + (−0.557 + 0.830i)10-s + (−0.117 − 0.993i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.184 − 0.982i)14-s + (0.0168 + 0.999i)15-s + (0.857 − 0.514i)16-s + (−0.250 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.415 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.335645221 - 1.501282773i\)
\(L(\frac12)\) \(\approx\) \(2.335645221 - 1.501282773i\)
\(L(1)\) \(\approx\) \(1.998342458 - 0.7417285840i\)
\(L(1)\) \(\approx\) \(1.998342458 - 0.7417285840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.990 - 0.134i)T \)
3 \( 1 + (0.736 - 0.676i)T \)
5 \( 1 + (-0.664 + 0.747i)T \)
7 \( 1 + (-0.0506 - 0.998i)T \)
11 \( 1 + (-0.117 - 0.993i)T \)
13 \( 1 + (0.918 + 0.394i)T \)
17 \( 1 + (-0.250 + 0.968i)T \)
19 \( 1 + (-0.874 + 0.485i)T \)
23 \( 1 + (-0.954 + 0.299i)T \)
29 \( 1 + (0.780 - 0.625i)T \)
31 \( 1 + (0.528 + 0.848i)T \)
37 \( 1 + (-0.972 - 0.234i)T \)
41 \( 1 + (0.347 + 0.937i)T \)
43 \( 1 + (0.963 - 0.266i)T \)
47 \( 1 + (0.943 - 0.331i)T \)
53 \( 1 + (0.0843 + 0.996i)T \)
59 \( 1 + (-0.931 - 0.363i)T \)
61 \( 1 + (0.736 - 0.676i)T \)
67 \( 1 + (0.979 + 0.201i)T \)
71 \( 1 + (-0.713 + 0.701i)T \)
73 \( 1 + (-0.905 - 0.425i)T \)
79 \( 1 + (-0.557 + 0.830i)T \)
83 \( 1 + (0.0843 + 0.996i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.250 + 0.968i)T \)
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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

−24.7772815525303451493531311984, −23.952156953240918665590374343536, −22.84896868214160485092586036776, −22.243430309030873017880003301850, −21.10384153309282684125420825611, −20.62580673123273694825740261792, −19.9029813914772822993785278628, −18.96020259356000859200772698405, −17.52602937775302232413336540109, −16.1520545053119585234155809662, −15.7042180192133888738108015109, −15.17245949169554449230219836520, −14.13068798067098603456748216326, −13.09895132206640390575686377781, −12.353598141912472539797014576517, −11.44080115170274705470616574903, −10.31220653568760266412273439607, −8.9977080961483685162003979648, −8.29372969021302215584227749633, −7.248654243648479299783957161244, −5.80639381028652023158339975637, −4.77377263972782532209040073971, −4.16820752036877437824652117518, −2.963380982717926452762483913025, −2.001109082495113241146751475124, 1.263737644870008823015383743035, 2.58565703020312061593241636852, 3.74207532497687100327716506734, 4.029830076338009520632252562163, 6.13726410282488282466804768559, 6.61660712244115210104851234788, 7.73687364202119717610394027395, 8.44940281117918905080610569551, 10.301801053725237225447283407948, 10.96272259365479590340590000733, 11.974356069295186833748973299302, 12.9513076651079091323110727053, 14.012492484640564036500416761349, 14.11560290596839758618232181553, 15.384944148461617175421354068921, 16.084249576094108268958625116999, 17.38257504263660332499518020448, 18.76673968936282837567457019109, 19.33039347480388696600740214604, 20.00129052925802765550220139535, 21.00842455026057265959203410771, 21.75769724398550985605157677571, 23.111238436709908514871127134793, 23.49910733304564415605734399677, 24.11826332106221777359041597081

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