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Properties

Label 1-193-193.106-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.707 + 0.707i$
Analytic cond. $20.7407$
Root an. cond. $20.7407$
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  + (−0.555 − 0.831i)2-s + (−0.923 − 0.382i)3-s + (−0.382 + 0.923i)4-s + (−0.881 − 0.471i)5-s + (0.195 + 0.980i)6-s + (0.707 + 0.707i)7-s + (0.980 − 0.195i)8-s + (0.707 + 0.707i)9-s + (0.0980 + 0.995i)10-s + (0.290 − 0.956i)11-s + (0.707 − 0.707i)12-s + (−0.995 − 0.0980i)13-s + (0.195 − 0.980i)14-s + (0.634 + 0.773i)15-s + (−0.707 − 0.707i)16-s + (0.881 − 0.471i)17-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (−0.923 − 0.382i)3-s + (−0.382 + 0.923i)4-s + (−0.881 − 0.471i)5-s + (0.195 + 0.980i)6-s + (0.707 + 0.707i)7-s + (0.980 − 0.195i)8-s + (0.707 + 0.707i)9-s + (0.0980 + 0.995i)10-s + (0.290 − 0.956i)11-s + (0.707 − 0.707i)12-s + (−0.995 − 0.0980i)13-s + (0.195 − 0.980i)14-s + (0.634 + 0.773i)15-s + (−0.707 − 0.707i)16-s + (0.881 − 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1073654750 - 0.2591926153i\)
\(L(\frac12)\) \(\approx\) \(-0.1073654750 - 0.2591926153i\)
\(L(1)\) \(\approx\) \(0.4076771883 - 0.2764785286i\)
\(L(1)\) \(\approx\) \(0.4076771883 - 0.2764785286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + (-0.555 - 0.831i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (-0.881 - 0.471i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.290 - 0.956i)T \)
13 \( 1 + (-0.995 - 0.0980i)T \)
17 \( 1 + (0.881 - 0.471i)T \)
19 \( 1 + (0.471 - 0.881i)T \)
23 \( 1 + (-0.555 - 0.831i)T \)
29 \( 1 + (0.773 + 0.634i)T \)
31 \( 1 + (-0.831 + 0.555i)T \)
37 \( 1 + (0.773 - 0.634i)T \)
41 \( 1 + (-0.290 + 0.956i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.0980 - 0.995i)T \)
53 \( 1 + (-0.995 - 0.0980i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (0.471 + 0.881i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (-0.471 + 0.881i)T \)
73 \( 1 + (-0.773 - 0.634i)T \)
79 \( 1 + (-0.956 - 0.290i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
89 \( 1 + (0.995 - 0.0980i)T \)
97 \( 1 + (-0.555 + 0.831i)T \)
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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

−27.39207222952147417521992550583, −26.730556944565085219001152218779, −25.68204777378357592185510915634, −24.318681137036801049335911671540, −23.59215212115135092122542085250, −22.9642085808078756100270577624, −22.09213536090265971844470085848, −20.5768193966254590796083582159, −19.54804189038105428857900927230, −18.46697644788266809054396661063, −17.50612128692669153260431929478, −16.89149058084043579309413612763, −15.86668980302603631360336722866, −14.913194338105251399225653734793, −14.26958272192957233968099058807, −12.36114331500581165327914110728, −11.41422647271273721984570325396, −10.32623485010512812050452978172, −9.65875425698019334306325647424, −7.76576944692975988047233931595, −7.39074217510191526703132598605, −6.100733094782372537943434727306, −4.81315997773960579946313047716, −4.008248808959110926784964412854, −1.362214795784259564262247252373, 0.163144137293995184135182314772, 1.24736629718688582187188360228, 2.8411116595100330798501929234, 4.468470017669235737814769207167, 5.376588489228160001769017333498, 7.19539481089431218497195471429, 8.09823469982005324197127057009, 9.11394933221597419106731806338, 10.55553017644300366700967149175, 11.57559684171606758908808100520, 11.99338425637987104764458479672, 12.863795117674884111520061859585, 14.33664613087825409982280110600, 16.00042383927411050654511606314, 16.64529337876278171866815702975, 17.72408802031743930700309516854, 18.5583734233346397959574977650, 19.35641788337019272378188946137, 20.308989667312290431993485976810, 21.63507471832564760444982001130, 22.09129295020546656015385531841, 23.35727308850559271544719348758, 24.29675780703331471424765402526, 25.06691438897206168635900383139, 26.87824845426814622708932359276

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