Properties

Label 1-373-373.139-r1-0-0
Degree $1$
Conductor $373$
Sign $0.742 - 0.669i$
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  + (0.988 + 0.151i)2-s + (0.0506 − 0.998i)3-s + (0.954 + 0.299i)4-s + (0.848 + 0.528i)5-s + (0.201 − 0.979i)6-s + (0.250 − 0.968i)7-s + (0.897 + 0.440i)8-s + (−0.994 − 0.101i)9-s + (0.758 + 0.651i)10-s + (0.897 − 0.440i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.394 − 0.918i)14-s + (0.571 − 0.820i)15-s + (0.820 + 0.571i)16-s + (0.954 + 0.299i)17-s + ⋯
L(s)  = 1  + (0.988 + 0.151i)2-s + (0.0506 − 0.998i)3-s + (0.954 + 0.299i)4-s + (0.848 + 0.528i)5-s + (0.201 − 0.979i)6-s + (0.250 − 0.968i)7-s + (0.897 + 0.440i)8-s + (−0.994 − 0.101i)9-s + (0.758 + 0.651i)10-s + (0.897 − 0.440i)11-s + (0.347 − 0.937i)12-s + (0.440 + 0.897i)13-s + (0.394 − 0.918i)14-s + (0.571 − 0.820i)15-s + (0.820 + 0.571i)16-s + (0.954 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.742 - 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.893404239 - 1.880714789i\)
\(L(\frac12)\) \(\approx\) \(4.893404239 - 1.880714789i\)
\(L(1)\) \(\approx\) \(2.488722426 - 0.5472246644i\)
\(L(1)\) \(\approx\) \(2.488722426 - 0.5472246644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.988 + 0.151i)T \)
3 \( 1 + (0.0506 - 0.998i)T \)
5 \( 1 + (0.848 + 0.528i)T \)
7 \( 1 + (0.250 - 0.968i)T \)
11 \( 1 + (0.897 - 0.440i)T \)
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 \( 1 + (0.688 - 0.724i)T \)
31 \( 1 + (-0.347 - 0.937i)T \)
37 \( 1 + (0.612 + 0.790i)T \)
41 \( 1 + (0.979 + 0.201i)T \)
43 \( 1 + (-0.299 + 0.954i)T \)
47 \( 1 + (-0.394 - 0.918i)T \)
53 \( 1 + (-0.101 - 0.994i)T \)
59 \( 1 + (-0.688 + 0.724i)T \)
61 \( 1 + (-0.998 - 0.0506i)T \)
67 \( 1 + (0.848 + 0.528i)T \)
71 \( 1 + (0.954 - 0.299i)T \)
73 \( 1 + (-0.994 + 0.101i)T \)
79 \( 1 + (0.651 - 0.758i)T \)
83 \( 1 + (-0.994 + 0.101i)T \)
89 \( 1 - T \)
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

−24.70080811975579943159126432410, −23.35505861462219793929584094782, −22.54464083979438402825125884130, −21.73847446135552264247181309416, −21.24143526585906439710976547733, −20.43731171555306073301067236532, −19.73274806771927000606517480902, −18.25510323248228024402399519211, −17.163513742254648630757148218544, −16.25550132287907267871568190165, −15.537569476425658155703065378628, −14.399415167719948029383611926873, −14.18108290951087598503031082541, −12.56720139628508107293198976527, −12.17363600065912879515149809260, −10.88069820745050069225297430691, −10.02956166671582147991509899063, −9.1391814480701103672976624053, −8.0430969825355178107958918707, −6.20380263879172860137299017925, −5.63319580415686572489848378672, −4.80634936801672132341934965067, −3.719692751908943591282893237159, −2.60777263567063561425620822875, −1.437429488942839648502456727193, 1.18159891890160738737924741504, 2.09324578663668619553590342176, 3.315527600604103576136393186507, 4.41059871117906804653378214838, 5.97580671465056789851648429986, 6.415330436800788017432731445614, 7.2915736468094851214293339468, 8.323958309842756711390234793587, 9.83723026033933419323106593329, 11.12055503559335566310076981747, 11.65186752884124554561269229013, 12.91205360623711971057602811800, 13.710030194078500048512515443570, 14.13452009346411305633838760527, 14.87590272487765421368796856756, 16.592011117124719845358676447235, 17.03022460396605263584592119209, 18.03790755810717783882631026373, 19.200777505633444929141091845845, 19.89915438597020589971712347017, 21.01242173352060049458273724135, 21.72630528029882225288142751038, 22.7278277880877648702432222514, 23.48820865506295316522784353259, 24.17013626874445779825143821521

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