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Properties

Label 1-145-145.82-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.190 + 0.981i$
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  + (−0.974 − 0.222i)2-s + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (−0.222 − 0.974i)6-s + (−0.433 − 0.900i)7-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (0.623 + 0.781i)11-s + i·12-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + i·17-s + (0.781 − 0.623i)18-s + (0.900 + 0.433i)19-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (−0.222 − 0.974i)6-s + (−0.433 − 0.900i)7-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (0.623 + 0.781i)11-s + i·12-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + i·17-s + (0.781 − 0.623i)18-s + (0.900 + 0.433i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.190 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6958855693 + 0.8442991778i\)
\(L(\frac12)\) \(\approx\) \(0.6958855693 + 0.8442991778i\)
\(L(1)\) \(\approx\) \(0.7602671311 + 0.2620100111i\)
\(L(1)\) \(\approx\) \(0.7602671311 + 0.2620100111i\)

Euler product

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

−27.7889077791811577796145459643, −26.47258257862760295755936373214, −25.81319462547981958470792163451, −24.722935809539653514221635576917, −24.37586804574378593049287224829, −23.03716326593348952721108787085, −21.60541655744438243306452460423, −20.3149273034065276358590475494, −19.50018875619362057625936765629, −18.49060226525755301225268154293, −18.1568400068685592677140658261, −16.6366982130943449343666045271, −15.78843422523303279554261480209, −14.5168122332560939113286438973, −13.50010505772582784827671885344, −11.96246440390141455435286964046, −11.37810581535835704263630530083, −9.47260648729551513356692925159, −8.89541692521739146529196363236, −7.796257857847529204850518788273, −6.57884846938348216710968652223, −5.81663014803699963293916577213, −3.22427798972024797632478343070, −2.014299266540536958845654212822, −0.57456980925722060235788858991, 1.41990180379932204908183172555, 3.19087656222460609524381929616, 4.1071890428824243324196136117, 6.074502702695784610442569975970, 7.4991514209587395876835318160, 8.50002410222155814057650958787, 9.74621759613963433622774707295, 10.27618759875371402400183140185, 11.36600015997302859185071217140, 12.79387003006294758961290920107, 14.199440066863726658382581851819, 15.395995152967302873330795712533, 16.275003105121653709561024672398, 17.12645261897739451061750595874, 18.178405388674629344052287409405, 19.770339512679180543479870593208, 19.985637771017040551497861873424, 21.01104689621778580802874407571, 22.12360829410938778277645966255, 23.21651870140002788124453696212, 24.79244216325586562102623849827, 25.76777363825689263493282209978, 26.28418188551140324752855271972, 27.33968358197514447410426642992, 28.002123269788202835881744795489

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