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Properties

Label 1-373-373.54-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.872 - 0.487i$
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  + (0.882 − 0.470i)2-s + (−0.943 + 0.331i)3-s + (0.557 − 0.830i)4-s + (0.455 − 0.890i)5-s + (−0.676 + 0.736i)6-s + (−0.918 − 0.394i)7-s + (0.101 − 0.994i)8-s + (0.780 − 0.625i)9-s + (−0.0168 − 0.999i)10-s + (0.810 − 0.585i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (−0.996 + 0.0843i)14-s + (−0.134 + 0.990i)15-s + (−0.378 − 0.925i)16-s + (0.440 + 0.897i)17-s + ⋯
L(s)  = 1  + (0.882 − 0.470i)2-s + (−0.943 + 0.331i)3-s + (0.557 − 0.830i)4-s + (0.455 − 0.890i)5-s + (−0.676 + 0.736i)6-s + (−0.918 − 0.394i)7-s + (0.101 − 0.994i)8-s + (0.780 − 0.625i)9-s + (−0.0168 − 0.999i)10-s + (0.810 − 0.585i)11-s + (−0.250 + 0.968i)12-s + (0.994 − 0.101i)13-s + (−0.996 + 0.0843i)14-s + (−0.134 + 0.990i)15-s + (−0.378 − 0.925i)16-s + (0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.872 - 0.487i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.872 - 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6270303613 - 2.407957043i\)
\(L(\frac12)\) \(\approx\) \(0.6270303613 - 2.407957043i\)
\(L(1)\) \(\approx\) \(1.178026787 - 0.8602645368i\)
\(L(1)\) \(\approx\) \(1.178026787 - 0.8602645368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.882 - 0.470i)T \)
3 \( 1 + (-0.943 + 0.331i)T \)
5 \( 1 + (0.455 - 0.890i)T \)
7 \( 1 + (-0.918 - 0.394i)T \)
11 \( 1 + (0.810 - 0.585i)T \)
13 \( 1 + (0.994 - 0.101i)T \)
17 \( 1 + (0.440 + 0.897i)T \)
19 \( 1 + (0.790 - 0.612i)T \)
23 \( 1 + (-0.651 - 0.758i)T \)
29 \( 1 + (0.638 - 0.769i)T \)
31 \( 1 + (0.250 + 0.968i)T \)
37 \( 1 + (0.315 + 0.948i)T \)
41 \( 1 + (-0.954 + 0.299i)T \)
43 \( 1 + (-0.830 - 0.557i)T \)
47 \( 1 + (-0.425 - 0.905i)T \)
53 \( 1 + (0.625 - 0.780i)T \)
59 \( 1 + (0.985 + 0.168i)T \)
61 \( 1 + (-0.331 - 0.943i)T \)
67 \( 1 + (-0.998 + 0.0506i)T \)
71 \( 1 + (-0.997 + 0.0675i)T \)
73 \( 1 + (-0.931 + 0.363i)T \)
79 \( 1 + (0.999 - 0.0168i)T \)
83 \( 1 + (0.780 + 0.625i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.897 + 0.440i)T \)
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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

−24.91029148348207499975093386705, −23.553515831280098614918226501818, −22.91032541277688908120503569426, −22.358452981842845984119446389878, −21.78100662434419792308502716697, −20.6841409477972097801458488628, −19.39650903631924606679046939176, −18.2921869097482986012397891143, −17.743953811903195992903250036064, −16.515606078547005532232749091480, −16.01045841379134229417318936690, −14.954312386133153711635391482315, −13.86763735774088838364541467301, −13.24491636982967328551878074624, −12.04899187774468204230098469577, −11.62485293308540917147513361211, −10.38966653993824871682701424883, −9.37346071317100000714602421943, −7.62930339655584655459011250102, −6.81088289268802216152613325035, −6.11558766591718358609976378978, −5.454805931641906330684331439637, −3.99619484838172030587300002497, −2.97355942141839321997230399564, −1.60271571277845433150748625479, 0.61192124815016417905308405995, 1.43124306866957080738498745674, 3.3474295202896229775981440971, 4.14471943824556736741459417037, 5.2062460347477667752838263389, 6.17087187403674800174008744726, 6.61855234546746522060895677559, 8.63053979910570092393297382418, 9.85774397759532976219355804928, 10.3716866738655034775713887594, 11.608037280234919777406372070, 12.23472758760278561598135688282, 13.239483867005063875934638828511, 13.76582182526322008965731486918, 15.20217576753713262915185265822, 16.268489952308698332748970751661, 16.52813542248265804347932872498, 17.73433622375190768465099166668, 18.93038112678453012523934620045, 19.92481741935452415097228396049, 20.68431272488615116789842073491, 21.67004572916929012805453200977, 22.12846667667067875024593076549, 23.13855933022750098511749212361, 23.74683139139025677885362975778

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