Properties

Label 1-145-145.137-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.928 - 0.371i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.623 − 0.781i)6-s + (−0.974 − 0.222i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.433 + 0.900i)11-s − 12-s + (−0.433 − 0.900i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (0.974 − 0.222i)19-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.222 − 0.974i)4-s + (−0.623 − 0.781i)6-s + (−0.974 − 0.222i)7-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.433 + 0.900i)11-s − 12-s + (−0.433 − 0.900i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (0.974 − 0.222i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.928 - 0.371i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.928 - 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2505271763 - 1.298746024i\)
\(L(\frac12)\) \(\approx\) \(0.2505271763 - 1.298746024i\)
\(L(1)\) \(\approx\) \(0.8132230300 - 0.9987166831i\)
\(L(1)\) \(\approx\) \(0.8132230300 - 0.9987166831i\)

Euler product

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

−28.68466620585223116536993257228, −27.23539491674843125378082191467, −26.64244741664858450183933826798, −25.652410681294698615360655282869, −24.93826291237060811021489667471, −23.641984730454711493048588234907, −22.636637912588473195288050595699, −21.82991605419907786405515923156, −21.19748597980126899390524456395, −19.839188409618709786194932097000, −18.709314136547892765771138072292, −16.978430115786888913270895686551, −16.41274081122341183921658826801, −15.628173850117218454620298014804, −14.44389075285303099307583384218, −13.781916731514245981958213037967, −12.38101057128992271420194734244, −11.27792334120987017695661561881, −9.60413461622818358679097096530, −8.940957890255382136610983428167, −7.4994387482735991316755255195, −6.126949379699394605043075173846, −5.18093496385836517864024955377, −3.76041961205846687889443215464, −3.01894055002309284282739810809, 0.982877324889039452588569328785, 2.55883023945689919465052141078, 3.5261233871742642398679141704, 5.24801382274950375541012083684, 6.46119044178789964575561472587, 7.53819775084230581917519112171, 9.26970511226691707915681818596, 10.15533523113914445643962853249, 11.65592247559666094314757335820, 12.5863082107363865489486524872, 13.150537659740161055769187402580, 14.32847803144160004042685464351, 15.21503331612324021625851155675, 16.87328661580517722216173485764, 18.15237954027024227654243526553, 18.99742966757628478248553417425, 20.01708752559522657964896310104, 20.43112961726212476282170429721, 22.14129100506655272696028774643, 22.84203831873148461606344013420, 23.593154325772776087837558851469, 24.843676664790489106172872069293, 25.50459957594022042808238581638, 26.94714878755080255265543940153, 28.2509713739722403571996631429

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