Properties

Label 1-145-145.94-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.626 - 0.779i$
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  + (0.900 − 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.623 − 0.781i)7-s + (0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s − 12-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.222 + 0.974i)18-s + (0.623 − 0.781i)19-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (−0.900 − 0.433i)6-s + (−0.623 − 0.781i)7-s + (0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s − 12-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.222 − 0.974i)16-s − 17-s + (0.222 + 0.974i)18-s + (0.623 − 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.626 - 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5780275297 - 1.206268032i\)
\(L(\frac12)\) \(\approx\) \(0.5780275297 - 1.206268032i\)
\(L(1)\) \(\approx\) \(1.003268987 - 0.8298831388i\)
\(L(1)\) \(\approx\) \(1.003268987 - 0.8298831388i\)

Euler product

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

−28.738547994900901084111931511386, −27.65620608137799100344624977176, −26.462516801383937424363241864284, −25.5220224043569431236525845026, −24.66766079122590266710223166712, −23.31380621172094236511621622611, −22.568878290537431997648791104, −22.06671329793102572892819572142, −20.84091649793055672927736971831, −20.13656137787540478255275507089, −18.26817447353347707374627517599, −17.28494556411898875961527866468, −16.15622787183080568071404786942, −15.419306035123768765543387043137, −14.75757546159995867177447318477, −13.06278239782988586949854633598, −12.34911735943481739291992309381, −11.23898700872476533803034018930, −10.033168505848463645564330282207, −8.722531400961035674185122766800, −7.11916162238180475211655488473, −5.935363577837206885743873222504, −5.13158239116512603718213360102, −3.89638832504576452669102963048, −2.63352361891419783460562572632, 0.988635556486008428616990518509, 2.603044850283146741371481246197, 4.04921104852478074822575052196, 5.40544717469767082433996817441, 6.53037255644626520688304446712, 7.28494712210351614923676688124, 9.2435316497444489704424173173, 10.92412149621726241961013264598, 11.25797596665690739832192897641, 12.69269759452069249577646306988, 13.43524898930687492343874983689, 14.132352509713761123908881845012, 15.88145629238719417711726229092, 16.58316876181107183056865131968, 17.98279060361113803141207408692, 19.24517743962957500608575404051, 19.69826944394446060141093931805, 21.14621907370020513586351527504, 22.13655784953939088944285839934, 22.96409107613474588033146630967, 23.902483913653775608154507123183, 24.37944502151712999361425318143, 25.73411123349006320798552769375, 26.984615453079427292691985583518, 28.5083945449258776589314278691

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