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Properties

Label 1-373-373.311-r1-0-0
Degree $1$
Conductor $373$
Sign $0.986 + 0.164i$
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  + (−0.625 + 0.780i)2-s + (0.839 + 0.543i)3-s + (−0.217 − 0.975i)4-s + (−0.882 + 0.470i)5-s + (−0.948 + 0.315i)6-s + (0.250 − 0.968i)7-s + (0.897 + 0.440i)8-s + (0.409 + 0.912i)9-s + (0.184 − 0.982i)10-s + (−0.830 − 0.557i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.598 + 0.801i)14-s + (−0.996 − 0.0843i)15-s + (−0.905 + 0.425i)16-s + (0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + (0.839 + 0.543i)3-s + (−0.217 − 0.975i)4-s + (−0.882 + 0.470i)5-s + (−0.948 + 0.315i)6-s + (0.250 − 0.968i)7-s + (0.897 + 0.440i)8-s + (0.409 + 0.912i)9-s + (0.184 − 0.982i)10-s + (−0.830 − 0.557i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.598 + 0.801i)14-s + (−0.996 − 0.0843i)15-s + (−0.905 + 0.425i)16-s + (0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.986 + 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286889052 + 0.1065396354i\)
\(L(\frac12)\) \(\approx\) \(1.286889052 + 0.1065396354i\)
\(L(1)\) \(\approx\) \(0.8239176691 + 0.3066579835i\)
\(L(1)\) \(\approx\) \(0.8239176691 + 0.3066579835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.625 + 0.780i)T \)
3 \( 1 + (0.839 + 0.543i)T \)
5 \( 1 + (-0.882 + 0.470i)T \)
7 \( 1 + (0.250 - 0.968i)T \)
11 \( 1 + (-0.830 - 0.557i)T \)
13 \( 1 + (0.440 + 0.897i)T \)
17 \( 1 + (0.954 + 0.299i)T \)
19 \( 1 + (-0.571 - 0.820i)T \)
23 \( 1 + (-0.998 + 0.0506i)T \)
29 \( 1 + (-0.972 - 0.234i)T \)
31 \( 1 + (-0.347 - 0.937i)T \)
37 \( 1 + (0.378 - 0.925i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (0.975 - 0.217i)T \)
47 \( 1 + (0.993 + 0.117i)T \)
53 \( 1 + (0.912 + 0.409i)T \)
59 \( 1 + (-0.283 - 0.959i)T \)
61 \( 1 + (0.543 - 0.839i)T \)
67 \( 1 + (0.848 + 0.528i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.585 + 0.810i)T \)
79 \( 1 + (-0.982 - 0.184i)T \)
83 \( 1 + (0.409 - 0.912i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.299 - 0.954i)T \)
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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

−24.57723420726888488725541382133, −23.51604597946284572663168292627, −22.627681589673353014255594045395, −21.24329768980661641781628766504, −20.66216834262302735064030529402, −20.0281805873728326574254237414, −19.040055331905581622379177143839, −18.457812402085026592368271891776, −17.79114333778976415616547717640, −16.34915484898456950814216517243, −15.517221941041049569977794275773, −14.59425491771484127921127941016, −13.20606752227705818310540516079, −12.414691954591031057512775988921, −12.060128582737530163963758225399, −10.711194366991896035804512930579, −9.630616105396794537517060180360, −8.61144298442046462189185502074, −8.02276562873146494974486868363, −7.403209143052972787173920907377, −5.571544962733591843374844706119, −4.11449319503481658994446535946, −3.13109965208816664914048691183, −2.15173294761073746655424980673, −0.97450244486841092061739257681, 0.494602038893392309555798692331, 2.193312247373511666111721795751, 3.76999391297983743261509538769, 4.41772927118633491660131067999, 5.87178754423140625397102960372, 7.30758561606846527487797415554, 7.75023093386334231837582664023, 8.619784749517152974618314199567, 9.69275062549052916402183592815, 10.70843090887530273081958326545, 11.18283127599665654181091460878, 13.143050036335834012761352927246, 14.119971712680076774180505558170, 14.63198369943219658088812814840, 15.70516314692318019640097304975, 16.213994920224885179000928346426, 17.10831040304966748546039235407, 18.51306216808911965987227202108, 19.03787715027860928431138501816, 19.8661758968787546118939432182, 20.68360896341624875880613389513, 21.75401072008400505582349300523, 23.01040393625877690523321724716, 23.783166788001363012811707543030, 24.27529465736011847429164887244

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