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Properties

Label 1-145-145.127-r0-0-0
Degree $1$
Conductor $145$
Sign $-0.679 + 0.734i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.679 + 0.734i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ -0.679 + 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6305830952 + 1.442252126i\)
\(L(\frac12)\) \(\approx\) \(0.6305830952 + 1.442252126i\)
\(L(1)\) \(\approx\) \(1.028023846 + 1.052911572i\)
\(L(1)\) \(\approx\) \(1.028023846 + 1.052911572i\)

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

−28.17247982518285245873559792112, −27.131663979625472704188658974368, −25.73821721124752099528712646594, −24.56330549125811741522618888902, −23.76101476508410915740697601825, −23.26014428481929504927715661137, −21.615843910030804445690908476073, −21.0690178017438960951784646032, −19.91508816930276148543707129638, −18.79366190063372639399004818214, −18.416813756042589012883622433765, −16.998784720915251779033219803947, −15.3225117746274136016798506559, −14.10880824718808789207578475115, −13.71633996382606264931202089796, −12.34191265317921808840184852734, −11.63160500865977751084379102821, −10.6123689654806258928769297978, −8.951922116215014407019728668410, −7.96727746240044326339263787973, −6.34952802021737144556405088901, −5.36439776887953451267871564604, −3.793290583127576453943588219907, −2.38902129092622124499046056091, −1.29499788792136773980683591853, 2.620694742378001164200746867245, 4.08462951205853870020614934832, 4.881450691793828598730375018801, 5.9869000046482701838563763818, 7.748629658781018197385039695093, 8.36123549643266075416058353564, 9.88080956100224071295635170335, 11.02529051683120778686645967006, 12.3402593762093414941217285692, 13.599339110338391031685016134124, 14.747054007163167241789154971069, 15.16071005703034577076583098712, 16.33653947311142403421823469875, 17.28003598384811120465571353397, 18.19761166580419769248196991988, 20.15716300728641340959742818520, 20.89400314970013836431348938167, 21.63413193302341786778900427976, 22.863821783013171098605148080124, 23.45223082161391286439571359794, 24.79972790359526090078111788278, 25.655575443305390135284151680, 26.447691855034622977305481140594, 27.53402603688303830771850066639, 28.14088596714089517862805088785

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