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Properties

Label 1-363-363.299-r0-0-0
Degree $1$
Conductor $363$
Sign $0.933 + 0.357i$
Analytic cond. $1.68576$
Root an. cond. $1.68576$
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  + (0.941 + 0.336i)2-s + (0.774 + 0.633i)4-s + (−0.974 − 0.226i)5-s + (0.736 − 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.841 − 0.540i)10-s + (0.998 + 0.0570i)13-s + (0.921 − 0.389i)14-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (0.985 + 0.170i)19-s + (−0.610 − 0.791i)20-s + (−0.415 + 0.909i)23-s + (0.897 + 0.441i)25-s + (0.921 + 0.389i)26-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (0.774 + 0.633i)4-s + (−0.974 − 0.226i)5-s + (0.736 − 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.841 − 0.540i)10-s + (0.998 + 0.0570i)13-s + (0.921 − 0.389i)14-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (0.985 + 0.170i)19-s + (−0.610 − 0.791i)20-s + (−0.415 + 0.909i)23-s + (0.897 + 0.441i)25-s + (0.921 + 0.389i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ 0.933 + 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.187582858 + 0.4049549664i\)
\(L(\frac12)\) \(\approx\) \(2.187582858 + 0.4049549664i\)
\(L(1)\) \(\approx\) \(1.736259556 + 0.2610196108i\)
\(L(1)\) \(\approx\) \(1.736259556 + 0.2610196108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.941 + 0.336i)T \)
5 \( 1 + (-0.974 - 0.226i)T \)
7 \( 1 + (0.736 - 0.676i)T \)
13 \( 1 + (0.998 + 0.0570i)T \)
17 \( 1 + (-0.466 - 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.897 - 0.441i)T \)
31 \( 1 + (-0.362 - 0.931i)T \)
37 \( 1 + (-0.254 + 0.967i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.564 - 0.825i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (0.0285 + 0.999i)T \)
61 \( 1 + (-0.941 + 0.336i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.696 - 0.717i)T \)
73 \( 1 + (-0.993 - 0.113i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (-0.870 - 0.491i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.974 - 0.226i)T \)
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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

−24.28563520233094843803190366431, −23.7971040104159621617250431367, −22.847966402287526818714394552233, −22.10694387561226986804601029719, −21.2064226146627158287894420242, −20.37392741491573614661927249867, −19.557298948681569606346438751130, −18.6555561728753357719323750043, −17.781245000738418454015845379029, −16.03803300346777553081221586013, −15.74420838201116906682731358591, −14.6625213171189547274391915048, −14.05370943057941016741591217282, −12.70783421558384015197465395524, −12.07322551191393597097621933958, −11.11572561341064886888164597094, −10.597226273672899883741394970886, −8.906821276228980660550555809115, −7.954729988770424452231968015098, −6.7944433979017095853754600648, −5.75134610758881036695486547770, −4.667269009434480396862314965472, −3.76541192374199015630435763587, −2.70725177530587906298678807804, −1.37160482752588827764563929641, 1.338286084898790860586568620865, 3.04059724031796885596097723519, 4.04805600772286216289654419449, 4.74794042410559285718489253770, 5.91647447872277639009991161229, 7.24894751043184548056692884887, 7.76616366755910934451849511127, 8.8189168607528082553880943326, 10.516189768149708127056682345975, 11.59383043128872705358789168995, 11.838983416291723251648669862593, 13.383978662947042520057763813813, 13.80838414798419875766045647873, 14.97830203518070987708932322702, 15.77210200714424154070925384161, 16.42585901987054018708502535419, 17.48140362210065288602643406855, 18.536735270535741105169214059487, 19.96046225628416409647585541991, 20.38371733458775811613239616701, 21.2106748303189612865093716928, 22.3886442658847499657805114971, 23.144364958610470627399825206133, 23.8013849919660309402316307064, 24.442058031142151906414978402306

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