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Properties

Label 1-373-373.160-r0-0-0
Degree $1$
Conductor $373$
Sign $0.963 + 0.267i$
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  + (−0.979 + 0.201i)2-s + (−0.440 + 0.897i)3-s + (0.918 − 0.394i)4-s + (0.954 + 0.299i)5-s + (0.250 − 0.968i)6-s + (−0.758 − 0.651i)7-s + (−0.820 + 0.571i)8-s + (−0.612 − 0.790i)9-s + (−0.994 − 0.101i)10-s + (−0.820 − 0.571i)11-s + (−0.0506 + 0.998i)12-s + (0.820 + 0.571i)13-s + (0.874 + 0.485i)14-s + (−0.688 + 0.724i)15-s + (0.688 − 0.724i)16-s + (0.918 − 0.394i)17-s + ⋯
L(s)  = 1  + (−0.979 + 0.201i)2-s + (−0.440 + 0.897i)3-s + (0.918 − 0.394i)4-s + (0.954 + 0.299i)5-s + (0.250 − 0.968i)6-s + (−0.758 − 0.651i)7-s + (−0.820 + 0.571i)8-s + (−0.612 − 0.790i)9-s + (−0.994 − 0.101i)10-s + (−0.820 − 0.571i)11-s + (−0.0506 + 0.998i)12-s + (0.820 + 0.571i)13-s + (0.874 + 0.485i)14-s + (−0.688 + 0.724i)15-s + (0.688 − 0.724i)16-s + (0.918 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.963 + 0.267i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.963 + 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7276257430 + 0.09920281905i\)
\(L(\frac12)\) \(\approx\) \(0.7276257430 + 0.09920281905i\)
\(L(1)\) \(\approx\) \(0.6598574573 + 0.1399955455i\)
\(L(1)\) \(\approx\) \(0.6598574573 + 0.1399955455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.979 + 0.201i)T \)
3 \( 1 + (-0.440 + 0.897i)T \)
5 \( 1 + (0.954 + 0.299i)T \)
7 \( 1 + (-0.758 - 0.651i)T \)
11 \( 1 + (-0.820 - 0.571i)T \)
13 \( 1 + (0.820 + 0.571i)T \)
17 \( 1 + (0.918 - 0.394i)T \)
19 \( 1 + (-0.688 - 0.724i)T \)
23 \( 1 + (0.440 + 0.897i)T \)
29 \( 1 + (0.528 - 0.848i)T \)
31 \( 1 + (-0.0506 - 0.998i)T \)
37 \( 1 + (0.347 - 0.937i)T \)
41 \( 1 + (-0.250 + 0.968i)T \)
43 \( 1 + (-0.918 + 0.394i)T \)
47 \( 1 + (0.874 - 0.485i)T \)
53 \( 1 + (0.612 - 0.790i)T \)
59 \( 1 + (0.528 - 0.848i)T \)
61 \( 1 + (0.440 - 0.897i)T \)
67 \( 1 + (0.954 + 0.299i)T \)
71 \( 1 + (0.918 + 0.394i)T \)
73 \( 1 + (-0.612 + 0.790i)T \)
79 \( 1 + (0.994 + 0.101i)T \)
83 \( 1 + (-0.612 + 0.790i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.918 + 0.394i)T \)
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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

−25.01854964343581411824150201661, −23.83199572898770126631542126373, −22.92126980731134470295194129909, −21.808588234622836487231012969794, −20.91897306093408431041692974652, −20.091278484534774931416082765466, −18.87541390256571686298342833059, −18.46839168488433078943618559459, −17.71447481231989870652536585729, −16.80203922148840176194216265219, −16.13421665512378442125585154698, −14.91097065717444241797609862540, −13.45435956085124946794665060535, −12.53367534347601652041367940573, −12.2707050472038954191726068123, −10.56094736782926801847242874677, −10.25338179827293089484771682781, −8.84259471416533168595069566789, −8.21865685407702272996933385910, −6.93453179626641162281160480265, −6.12136727343817263233370741983, −5.35843264928507468270329951663, −3.065594216714445612394698212084, −2.148374753266864683625204050034, −1.08457020464796024711616098235, 0.78253485843661945421931871537, 2.53563926421588050522928662262, 3.60563539926226400464261033926, 5.292258263018811633254290780874, 6.12051885527850310525524445136, 6.89759995600352911427229141616, 8.3137030769992675125972344804, 9.48736661463534726511331225294, 9.88603640554748344305935803592, 10.83364812561649333421145742304, 11.42718773768127876157465630259, 13.08229098148326789760889445093, 14.04342624555927122102293545803, 15.19542310477259114830391759147, 16.07029527651346629696506755288, 16.72435240703595925962709118506, 17.40849772071403191172581857462, 18.38683983812179216907376884066, 19.18477887061348020712942690770, 20.37829617266855365459880830615, 21.181361236602411630687351051107, 21.68823636076403154246899314412, 23.16353757231078734810813658247, 23.55213716539495809346597538828, 25.0361017136901522438291623276

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