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Properties

Label 1-145-145.23-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: $\chi_{145} (23, \cdot )$
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

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

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