Properties

Label 1-145-145.7-r1-0-0
Degree $1$
Conductor $145$
Sign $0.123 + 0.992i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (−0.433 − 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.781 + 0.623i)7-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s + i·12-s + (−0.974 + 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.974 + 0.222i)18-s + (−0.623 + 0.781i)19-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.781 + 0.623i)7-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s + i·12-s + (−0.974 + 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s + i·17-s + (−0.974 + 0.222i)18-s + (−0.623 + 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.123 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1890081484 + 0.1669960919i\)
\(L(\frac12)\) \(\approx\) \(0.1890081484 + 0.1669960919i\)
\(L(1)\) \(\approx\) \(0.6508191831 - 0.3112127465i\)
\(L(1)\) \(\approx\) \(0.6508191831 - 0.3112127465i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.433 - 0.900i)T \)
3 \( 1 + (0.781 - 0.623i)T \)
7 \( 1 + (-0.781 + 0.623i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.433 + 0.900i)T \)
59 \( 1 - T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (-0.974 - 0.222i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.433 + 0.900i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 + (-0.781 - 0.623i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−27.4536470845153697728686718609, −26.57363080137091408507282775423, −25.88062897379115958233094727265, −25.138031840451236269942530592357, −24.05717221108220148969446798334, −22.79765381587464784211323019625, −22.131729465216066219824302641609, −20.454150550352548298255953168014, −19.79999871741441342113395005778, −18.80400980541642687955162034109, −17.4789003465797377159348022078, −16.512771874943605762875569187164, −15.61896534372621734771827966182, −14.75810991541117832459915498996, −13.7977081356900299789906079741, −12.734117567899450114481899451327, −10.57763561929715407752336984477, −9.81635558003074951077884755710, −9.00246478308183347051537555721, −7.61175741072332036467921381857, −6.89050534152544028485738604161, −5.11040137617626869615620548116, −4.14324720915853818742957819560, −2.41305338475437529228214106490, −0.09619588330566577216216828386, 1.70652966999974685300065226227, 2.85206036880658674085457007874, 3.84482844133644612484079906692, 5.90836100789308328584377195476, 7.448545267092609604512789583513, 8.531541323592020910226145462960, 9.34300947872604025654263793408, 10.47876478614116780854445583846, 12.0035806074250897028434084808, 12.68290742536809632593902295354, 13.64066323055429225950784056187, 14.79567521476033811347086372464, 16.26235086233104748655718975017, 17.47735009943574953875011756185, 18.66142489574779726405027493916, 19.25982686517528269485896947214, 19.89555495392606994709434664769, 21.29364882148396915512458320845, 21.8475333716267639739992132621, 23.24879695448473029455490250674, 24.445349421310448264002166871839, 25.545869773416972321903334708729, 26.25851885131573510856183927835, 27.19706447305213694319541198579, 28.41064131249836410663697600027

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