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Properties

Label 1-373-373.122-r0-0-0
Degree $1$
Conductor $373$
Sign $0.728 - 0.685i$
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  + (−0.780 + 0.625i)2-s + (−0.839 + 0.543i)3-s + (0.217 − 0.975i)4-s + (−0.470 + 0.882i)5-s + (0.315 − 0.948i)6-s + (−0.250 − 0.968i)7-s + (0.440 + 0.897i)8-s + (0.409 − 0.912i)9-s + (−0.184 − 0.982i)10-s + (0.557 + 0.830i)11-s + (0.347 + 0.937i)12-s + (−0.440 + 0.897i)13-s + (0.801 + 0.598i)14-s + (−0.0843 − 0.996i)15-s + (−0.905 − 0.425i)16-s + (−0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (−0.780 + 0.625i)2-s + (−0.839 + 0.543i)3-s + (0.217 − 0.975i)4-s + (−0.470 + 0.882i)5-s + (0.315 − 0.948i)6-s + (−0.250 − 0.968i)7-s + (0.440 + 0.897i)8-s + (0.409 − 0.912i)9-s + (−0.184 − 0.982i)10-s + (0.557 + 0.830i)11-s + (0.347 + 0.937i)12-s + (−0.440 + 0.897i)13-s + (0.801 + 0.598i)14-s + (−0.0843 − 0.996i)15-s + (−0.905 − 0.425i)16-s + (−0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.728 - 0.685i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.728 - 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2223107283 - 0.08810099236i\)
\(L(\frac12)\) \(\approx\) \(0.2223107283 - 0.08810099236i\)
\(L(1)\) \(\approx\) \(0.3985887401 + 0.1514641276i\)
\(L(1)\) \(\approx\) \(0.3985887401 + 0.1514641276i\)

Euler product

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

−24.85724691435871202198926191279, −24.04115095023231907010365702471, −22.78218387964420771821801849644, −22.01721827170835720619421919641, −21.268842019012514030353345031808, −20.049483774319725945587859334464, −19.31973527564789064841752224067, −18.69194083999924489311352648560, −17.62574981930265417068776113402, −17.00392820828786217892055633918, −16.125548987007042293306105686424, −15.36929057293005552284416407276, −13.39586001871673390556969838434, −12.72767033239341858744619952073, −11.91913176125263442593014501552, −11.397634657240179873223877043679, −10.2646737883631439203100636340, −9.029683202418003408490686748220, −8.377252685874571264971411553481, −7.349300436093969578564751143239, −6.12439415750487893478459151044, −5.09063080268139852196894087926, −3.72999452655739466814319967867, −2.30840603531226572325528039179, −1.10409877188556878481579353100, 0.24030718907616448946143623813, 2.07866128105268917411315699452, 4.03838915410628349073476724015, 4.65582481245263113298889230264, 6.45558503578860137972739728245, 6.6706014835788736753945346517, 7.63910035342133284320720662113, 9.17731769446612329434974695454, 9.9084486906305707310336520894, 10.92487105128621563772120912782, 11.26042802588479236154292906876, 12.69181388545861088473644693754, 14.27435228332490207704519638927, 14.90154220920089673918200697618, 15.76019053760289438418956988596, 16.691199725354040005913329477925, 17.32044122843253782437408916573, 18.06500057338699876847683922018, 19.187557078434917781047457223573, 19.83332178780024229238111342256, 20.93094968503630036390240935366, 22.400384868880446413243351900800, 22.69878571276345007327467642140, 23.73086768783108782864404389891, 24.25774879176916388069241748920

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