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Properties

Label 1-145-145.122-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.999 + 0.0361i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.999 + 0.0361i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.999 + 0.0361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01035044576 - 0.5725554975i\)
\(L(\frac12)\) \(\approx\) \(0.01035044576 - 0.5725554975i\)
\(L(1)\) \(\approx\) \(0.5976190899 - 0.2881222658i\)
\(L(1)\) \(\approx\) \(0.5976190899 - 0.2881222658i\)

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

−28.31233547684304480631696120182, −27.43904849711034114480777036622, −26.63161704743781357999589385568, −25.85079524993776239406250844850, −24.874667973934196106547694376533, −23.87554655244897094212463058513, −21.88475439867291406558032629292, −21.49988977494415401497232525639, −20.58570358396242997948240229450, −19.319111802036015087094049474529, −18.775504116658714599079179448488, −17.303114022456846637973097189609, −16.53007423265271155922795848112, −15.34784499945094424216982992866, −14.76312135128429565430752628046, −13.02003024557570284253869611609, −11.582435640642001900565809678319, −10.78147179886017632932997167356, −9.65276386373894738034423535186, −8.68185770866406025037294250336, −7.99417103751579208028891356563, −6.23396733158389365344206318376, −4.76726594614550268256898877492, −3.10344058927041558693532361669, −2.07052955006176531130355370908, 0.26760307817205303947671815588, 1.654680212100160258378494490314, 2.91139216988145659114484192795, 5.089025690660812980557121168635, 6.78456958474670529330242857655, 7.46932824744457050910721830082, 8.32123589962721498008081967107, 9.66329876097414521830262500773, 10.70239732860940744571346453046, 11.975887982377617568658100135901, 13.117037450835713891571931979719, 14.43140910458463642761625795365, 15.23425913813666947206917002682, 16.761126406313186687523407810132, 17.61529524312303258278377390246, 18.32539375881390570734635451145, 19.47536531785708894067687287137, 20.22916694954674984621746111322, 20.98556680435795451454065096839, 23.10301047081505230901966549367, 23.677895243448736787950738917583, 24.92323340010414973446877513817, 25.36845219707966211660291678204, 26.60359390247773355454098558989, 27.217080555684177547080198362578

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