Properties

Label 1-193-193.104-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.805 - 0.592i$
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  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (−0.0980 + 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (−0.290 + 0.956i)10-s + (−0.773 − 0.634i)11-s + (−0.707 + 0.707i)12-s + (−0.956 + 0.290i)13-s + (−0.555 − 0.831i)14-s + (−0.881 − 0.471i)15-s + (0.707 + 0.707i)16-s + (0.0980 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (−0.0980 + 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (−0.290 + 0.956i)10-s + (−0.773 − 0.634i)11-s + (−0.707 + 0.707i)12-s + (−0.956 + 0.290i)13-s + (−0.555 − 0.831i)14-s + (−0.881 − 0.471i)15-s + (0.707 + 0.707i)16-s + (0.0980 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2984908036 + 0.9089094429i\)
\(L(\frac12)\) \(\approx\) \(-0.2984908036 + 0.9089094429i\)
\(L(1)\) \(\approx\) \(0.9525197868 + 0.6811383903i\)
\(L(1)\) \(\approx\) \(0.9525197868 + 0.6811383903i\)

Euler product

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

−25.60895085769563870774631838108, −25.0197747461584635386057178658, −24.29012335194581062093605727164, −23.29621176029394648030771879034, −22.70391114840659654145325001282, −21.57992026641237961563246692190, −20.51643611800039892992906996459, −19.614781021273222106933280855815, −18.80664868622388427831225578764, −17.4452811423735313761573315143, −16.38431299350438630695762304671, −15.52470827854925652171983984607, −14.28642304544564684189673678329, −12.8549333647298751743050888429, −12.74478281203075419183458785173, −11.90696025761676198597135113454, −10.58024710371510857343599745833, −9.15898738737062000263932071028, −7.67423786164002973998764415254, −6.7123773504179489032132520871, −5.4024718429132374188134067990, −4.87662568904913329348296651288, −2.93099061531307019660138049095, −1.896096523629936256842490642503, −0.223860552760128035962687211968, 2.65613603942167103173204475949, 3.5786508191605689425409418809, 4.56401159067398330193451183068, 5.925358300955184310278910537989, 6.69971874449615413502670998183, 7.953926324120837812625791598804, 9.84836250951785363719523383053, 10.691127310919233390544622852154, 11.387920887894137925078473036666, 12.75973130508134331043316118140, 13.79430637172758502040537680430, 14.9505488475633173914429119642, 15.3827165887684876255059869712, 16.68618760071411910658247771030, 17.16969392302198885978354108936, 18.99277566643494770992390639722, 19.8768808702370786907226489709, 21.231244488380042118360804579335, 21.683004138599170660358911219508, 22.74068835166800028920628402171, 23.21227008891130556955656074864, 24.19798232726154782183699798417, 25.83543795568845210344012285549, 26.23951941307815465626384770542, 27.060997744530302620678236489206

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