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Properties

Label 1-193-193.13-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: $\chi_{193} (13, \cdot )$
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

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

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