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Properties

Label 1-2261-2261.109-r0-0-0
Degree $1$
Conductor $2261$
Sign $-0.379 - 0.925i$
Analytic cond. $10.5000$
Root an. cond. $10.5000$
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  + (−0.573 + 0.819i)2-s + (−0.843 + 0.537i)3-s + (−0.342 − 0.939i)4-s + (0.999 − 0.0436i)5-s + (0.0436 − 0.999i)6-s + (0.965 + 0.258i)8-s + (0.422 − 0.906i)9-s + (−0.537 + 0.843i)10-s + (0.608 + 0.793i)11-s + (0.793 + 0.608i)12-s + (−0.342 − 0.939i)13-s + (−0.819 + 0.573i)15-s + (−0.766 + 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.573 + 0.819i)2-s + (−0.843 + 0.537i)3-s + (−0.342 − 0.939i)4-s + (0.999 − 0.0436i)5-s + (0.0436 − 0.999i)6-s + (0.965 + 0.258i)8-s + (0.422 − 0.906i)9-s + (−0.537 + 0.843i)10-s + (0.608 + 0.793i)11-s + (0.793 + 0.608i)12-s + (−0.342 − 0.939i)13-s + (−0.819 + 0.573i)15-s + (−0.766 + 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2261\)    =    \(7 \cdot 17 \cdot 19\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(10.5000\)
Root analytic conductor: \(10.5000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2261} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2261,\ (0:\ ),\ -0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02691030129 - 0.04012611146i\)
\(L(\frac12)\) \(\approx\) \(0.02691030129 - 0.04012611146i\)
\(L(1)\) \(\approx\) \(0.5419149783 + 0.2389406627i\)
\(L(1)\) \(\approx\) \(0.5419149783 + 0.2389406627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.573 + 0.819i)T \)
3 \( 1 + (-0.843 + 0.537i)T \)
5 \( 1 + (0.999 - 0.0436i)T \)
11 \( 1 + (0.608 + 0.793i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.887 + 0.461i)T \)
29 \( 1 + (-0.953 - 0.300i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (-0.737 - 0.675i)T \)
43 \( 1 + (-0.819 - 0.573i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (-0.906 - 0.422i)T \)
59 \( 1 + (0.906 - 0.422i)T \)
61 \( 1 + (0.461 + 0.887i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.843 + 0.537i)T \)
73 \( 1 + (-0.537 - 0.843i)T \)
79 \( 1 + (0.300 + 0.953i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (-0.461 + 0.887i)T \)
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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

−19.77300828187145547761585347130, −18.93218406688719025641368916059, −18.55108725651293330659663579796, −17.79885841220139908166189697153, −17.18571830885919083386349162335, −16.579446111781433500725346363673, −16.158873415314783833683454733921, −14.51164025301243217181761568306, −13.85256887348907420308685638099, −13.24795370915390163171211506695, −12.46745798184391534210395379839, −11.774886937166815370112452030774, −11.19816336516342667456700131754, −10.3952540860338384288206060688, −9.79707366315747446934217061181, −8.931482459618451975716448335127, −8.26881359930539708379934821834, −7.06496437621697106499116859372, −6.62605286112475548004557900152, −5.65496273295792272896656488683, −4.83679762867990178488367831461, −3.84991738645967799374568189222, −2.76505293770598280221546617548, −1.7414490804961935335984182207, −1.41129237840118443062332418589, 0.02187479206999234105260566422, 1.31995717772765008699272967710, 2.12153156212485681668824523132, 3.68824864588034592745930488529, 4.61990826958498220934970099783, 5.477178009776505385588160721593, 5.79158977852099752908764361016, 6.72810931615316377506115365278, 7.32867101940175183209838191478, 8.416482320432446398012991452567, 9.40849354544603619370183967779, 9.77665485418808252374723457331, 10.3632600420137010301042779467, 11.15743296611017499217969340827, 12.157670523603966612370681644590, 12.96847368766834904373677738766, 13.820570405928431275718141294786, 14.71440505614966983474131956487, 15.19838335006908969430432193574, 15.95367809311258313037515227288, 16.81878397219643566981283719361, 17.29066723432739583633137285709, 17.73171142628820625051974899747, 18.34267820625771233798145365534, 19.24667923088673730873618242834

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