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Properties

Label 1-945-945.187-r0-0-0
Degree $1$
Conductor $945$
Sign $0.753 + 0.657i$
Analytic cond. $4.38856$
Root an. cond. $4.38856$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.866 + 0.5i)8-s + (−0.939 − 0.342i)11-s + (−0.342 − 0.939i)13-s + (0.766 − 0.642i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.984 + 0.173i)22-s + (−0.984 − 0.173i)23-s + (0.5 + 0.866i)26-s + (0.939 + 0.342i)29-s + (0.939 − 0.342i)31-s + (−0.642 + 0.766i)32-s + (0.766 − 0.642i)34-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.866 + 0.5i)8-s + (−0.939 − 0.342i)11-s + (−0.342 − 0.939i)13-s + (0.766 − 0.642i)16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.984 + 0.173i)22-s + (−0.984 − 0.173i)23-s + (0.5 + 0.866i)26-s + (0.939 + 0.342i)29-s + (0.939 − 0.342i)31-s + (−0.642 + 0.766i)32-s + (0.766 − 0.642i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(4.38856\)
Root analytic conductor: \(4.38856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 945,\ (0:\ ),\ 0.753 + 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6489687124 + 0.2434264939i\)
\(L(\frac12)\) \(\approx\) \(0.6489687124 + 0.2434264939i\)
\(L(1)\) \(\approx\) \(0.6278115093 + 0.05972455820i\)
\(L(1)\) \(\approx\) \(0.6278115093 + 0.05972455820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−21.34715184256590241859668668831, −21.0141100063005901870255352897, −19.79271386449675405140890679953, −19.53075887406239753451138210551, −18.46092284562646347579254183942, −17.8046800790566697829927580528, −17.248459977319910038031200665843, −15.99268476150578698603344429066, −15.84383447343699908791050810253, −14.72882065278876757411677627010, −13.67402198837213461163418019261, −12.7388291552696984850951996398, −11.86216190197174873382576426787, −11.14083634305969303751916222628, −10.285935049965250206254800813888, −9.530677142197305442579219417602, −8.72572308812135547205467034047, −7.882531868052290980340347275018, −7.0013555221779066895442099346, −6.32292233913304051228341871517, −5.00259791067323734357855532661, −4.003122367813112846950428404010, −2.535561039048691653490284333718, −2.15018915172936936588647610081, −0.55004732434476956097560590130, 0.83637447722823346787547159668, 2.19970260441085368029783024075, 2.91436418360556978671248778113, 4.31407761751140852695930993445, 5.59868462143648386873766046939, 6.207006281785965236806288301704, 7.30311678959964138426592090393, 8.19523224373802901956040600240, 8.565086348515056598006079729544, 9.94319594575775595693577578554, 10.3401738574957869788321397402, 11.15603424045605781500452995420, 12.193578797184232451298881351332, 12.95058911731896921329507697091, 14.08249363634283754200949027655, 15.020186072393941195739761284202, 15.73495268273528608993485509059, 16.32277566690796793927439032590, 17.45116254809969767253234754877, 17.78868548118669694689901164899, 18.76082299093750583243283921007, 19.38633263978409355334459504889, 20.25331349554770668102149133260, 20.87141855638540473177458915432, 21.742890768095874364378728355505

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