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Properties

Label 1-45-45.13-r1-0-0
Degree $1$
Conductor $45$
Sign $0.370 + 0.929i$
Analytic cond. $4.83592$
Root an. cond. $4.83592$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s − 19-s + (−0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 26-s + i·28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − i·17-s − 19-s + (−0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 26-s + i·28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.370 + 0.929i$
Analytic conductor: \(4.83592\)
Root analytic conductor: \(4.83592\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 45,\ (1:\ ),\ 0.370 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.089393876 + 1.416821558i\)
\(L(\frac12)\) \(\approx\) \(2.089393876 + 1.416821558i\)
\(L(1)\) \(\approx\) \(1.662104067 + 0.7152142400i\)
\(L(1)\) \(\approx\) \(1.662104067 + 0.7152142400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 - T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−33.563139373540307888599398043508, −32.64549932824481790078654114306, −31.3599118117517249822078823486, −30.46814483438549901161605320880, −29.42454792684348963400871208636, −28.2476323743655423292087950918, −27.01772544332566706321391337961, −25.41581901329055039096638944902, −23.84360188683474264936280512574, −23.47909827477132417934059145050, −21.643856442365695998624218737538, −21.03974624263197048446988086363, −19.67249252540702479960660775924, −18.43785434689263770344974137147, −16.69247618605106810511861226911, −15.20302890355786756304450356509, −14.01542558368522215548520337583, −12.96639624232198349239188900447, −11.3261798640330158092720241250, −10.54877009726074926044277444140, −8.456015699452004513454125940380, −6.5293187531311927236861314421, −4.98797822013136312246649457280, −3.52118562865846980276665911532, −1.49571308205630303629853667443, 2.430404657581143479635392788174, 4.41542201283699230114209826680, 5.671260819630638130284355295785, 7.33113221226391634339523192714, 8.637184166945910819260063887699, 10.87606403167583598028331885299, 12.19459817744109971385172785361, 13.425728737179964487181909763810, 14.82456003075162729769204335935, 15.66433874171719306735619303876, 17.22403862792126177208631829324, 18.38262166361494488055232062814, 20.48414765842990332426277294512, 21.184197901919040255671317947, 22.645703368796494131975320336025, 23.564101765526483804325748135303, 24.857456588195913533412338521291, 25.664790105262540115050056735206, 27.17809141484929892114680498560, 28.52881146468825999469346907028, 30.0733115480033484527436801700, 30.94146294318809681585321290365, 31.89120861406081866214211640974, 33.25653190038167112890619229099, 34.01172251291200690580101245801

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