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Properties

Label 1-373-373.136-r1-0-0
Degree $1$
Conductor $373$
Sign $0.998 - 0.0461i$
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  + (−0.571 + 0.820i)2-s + (−0.979 − 0.201i)3-s + (−0.347 − 0.937i)4-s + (−0.790 − 0.612i)5-s + (0.724 − 0.688i)6-s + (−0.528 − 0.848i)7-s + (0.968 + 0.250i)8-s + (0.918 + 0.394i)9-s + (0.954 − 0.299i)10-s + (0.968 − 0.250i)11-s + (0.151 + 0.988i)12-s + (0.250 + 0.968i)13-s + (0.998 + 0.0506i)14-s + (0.651 + 0.758i)15-s + (−0.758 + 0.651i)16-s + (−0.347 − 0.937i)17-s + ⋯
L(s)  = 1  + (−0.571 + 0.820i)2-s + (−0.979 − 0.201i)3-s + (−0.347 − 0.937i)4-s + (−0.790 − 0.612i)5-s + (0.724 − 0.688i)6-s + (−0.528 − 0.848i)7-s + (0.968 + 0.250i)8-s + (0.918 + 0.394i)9-s + (0.954 − 0.299i)10-s + (0.968 − 0.250i)11-s + (0.151 + 0.988i)12-s + (0.250 + 0.968i)13-s + (0.998 + 0.0506i)14-s + (0.651 + 0.758i)15-s + (−0.758 + 0.651i)16-s + (−0.347 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.998 - 0.0461i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.998 - 0.0461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5966839740 + 0.01377286234i\)
\(L(\frac12)\) \(\approx\) \(0.5966839740 + 0.01377286234i\)
\(L(1)\) \(\approx\) \(0.4936145194 + 0.04146789905i\)
\(L(1)\) \(\approx\) \(0.4936145194 + 0.04146789905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.571 + 0.820i)T \)
3 \( 1 + (-0.979 - 0.201i)T \)
5 \( 1 + (-0.790 - 0.612i)T \)
7 \( 1 + (-0.528 - 0.848i)T \)
11 \( 1 + (0.968 - 0.250i)T \)
13 \( 1 + (0.250 + 0.968i)T \)
17 \( 1 + (-0.347 - 0.937i)T \)
19 \( 1 + (-0.651 + 0.758i)T \)
23 \( 1 + (-0.201 - 0.979i)T \)
29 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (-0.151 + 0.988i)T \)
37 \( 1 + (0.874 + 0.485i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (0.937 - 0.347i)T \)
47 \( 1 + (-0.998 + 0.0506i)T \)
53 \( 1 + (0.394 + 0.918i)T \)
59 \( 1 + (0.994 - 0.101i)T \)
61 \( 1 + (-0.201 + 0.979i)T \)
67 \( 1 + (-0.790 - 0.612i)T \)
71 \( 1 + (-0.347 + 0.937i)T \)
73 \( 1 + (0.918 - 0.394i)T \)
79 \( 1 + (-0.299 - 0.954i)T \)
83 \( 1 + (0.918 - 0.394i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.937 + 0.347i)T \)
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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

−24.31154321596613793869853065664, −23.15101293729164567574298728697, −22.30270411754963913391283442013, −22.09400550129626096645670009634, −20.991636396797028255379441122463, −19.64108590616091131925028835871, −19.28087693866644346925906263725, −18.17983835329816048070717877151, −17.590147932540919405170096981839, −16.60874864628093859775236029739, −15.592537836087057415162927619578, −14.933840689350436153493798224771, −13.03700639863597516525940716012, −12.48954224284227519847648540598, −11.43402665779239786000121898805, −11.0692767182901359756561256777, −9.94663025166424435580480010917, −9.09093965896149663572656774006, −7.87987248541692351585984511143, −6.79503139010023350493309000406, −5.77912980292102158043547837339, −4.22180407493688563799354031158, −3.51507547900350236087022316278, −2.12599163002879353501919212671, −0.544525649583795852416781729031, 0.51367432901455480774400457361, 1.43786658702479355734733438361, 4.05760630244559368533002661125, 4.61079755949753942977224953611, 6.02368556689082805207543100466, 6.78987981977686322892801665216, 7.52587264585996446042410386311, 8.74864023673402136417724727470, 9.6472504705266496793852571613, 10.80188434293402029169852487513, 11.570047672396392771923323535031, 12.66798808042042273264447654222, 13.69821274448602456401583584083, 14.72969380480153420334097691979, 16.09061549342711588592228581890, 16.49021098215382241868040487591, 16.92524205408168332981113779047, 18.10388645657766182347804026764, 19.026787027507049102250772945903, 19.652392261414201968258267217358, 20.7120735596758033795100361144, 22.24527989813272633676823962317, 22.966109776227987474384222094012, 23.57566399629166100333837845788, 24.330733801179835202572126217178

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