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Properties

Label 1-51e2-2601.1546-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.348 - 0.937i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (0.969 − 0.243i)2-s + (0.881 − 0.473i)4-s + (0.952 − 0.303i)5-s + (−0.332 − 0.943i)7-s + (0.739 − 0.673i)8-s + (0.850 − 0.526i)10-s + (0.0307 − 0.999i)11-s + (0.213 + 0.976i)13-s + (−0.552 − 0.833i)14-s + (0.552 − 0.833i)16-s + (−0.273 − 0.961i)19-s + (0.696 − 0.717i)20-s + (−0.213 − 0.976i)22-s + (−0.650 + 0.759i)23-s + (0.816 − 0.577i)25-s + (0.445 + 0.895i)26-s + ⋯
L(s)  = 1  + (0.969 − 0.243i)2-s + (0.881 − 0.473i)4-s + (0.952 − 0.303i)5-s + (−0.332 − 0.943i)7-s + (0.739 − 0.673i)8-s + (0.850 − 0.526i)10-s + (0.0307 − 0.999i)11-s + (0.213 + 0.976i)13-s + (−0.552 − 0.833i)14-s + (0.552 − 0.833i)16-s + (−0.273 − 0.961i)19-s + (0.696 − 0.717i)20-s + (−0.213 − 0.976i)22-s + (−0.650 + 0.759i)23-s + (0.816 − 0.577i)25-s + (0.445 + 0.895i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (1546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.111994869 - 3.037666179i\)
\(L(\frac12)\) \(\approx\) \(2.111994869 - 3.037666179i\)
\(L(1)\) \(\approx\) \(1.926605930 - 0.9975349111i\)
\(L(1)\) \(\approx\) \(1.926605930 - 0.9975349111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.969 - 0.243i)T \)
5 \( 1 + (0.952 - 0.303i)T \)
7 \( 1 + (-0.332 - 0.943i)T \)
11 \( 1 + (0.0307 - 0.999i)T \)
13 \( 1 + (0.213 + 0.976i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (-0.650 + 0.759i)T \)
29 \( 1 + (0.0307 - 0.999i)T \)
31 \( 1 + (-0.213 - 0.976i)T \)
37 \( 1 + (0.602 + 0.798i)T \)
41 \( 1 + (-0.816 - 0.577i)T \)
43 \( 1 + (-0.998 + 0.0615i)T \)
47 \( 1 + (0.650 + 0.759i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (-0.153 + 0.988i)T \)
61 \( 1 + (0.153 + 0.988i)T \)
67 \( 1 + (0.969 + 0.243i)T \)
71 \( 1 + (0.982 + 0.183i)T \)
73 \( 1 + (-0.445 - 0.895i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (0.816 - 0.577i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (-0.332 - 0.943i)T \)
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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

−19.994309842624405692745157514895, −18.6598634154245711974156675848, −18.16972597256323143283660541866, −17.427424020448339805411825808505, −16.6301045011617778337073644164, −15.87966317448871450721809469835, −15.16091189055528773037456494643, −14.57678730888809597941765074254, −14.04388993338617997685983939791, −12.96383871199582265971784762180, −12.627776167251434536938347741913, −12.067061957612989503392092767501, −10.92645358332538397349800051859, −10.2882629767301476777252341138, −9.58505142750668621645136025439, −8.53333397061822941533732320971, −7.820961369375503889838809282823, −6.64585652077106711721519275289, −6.393071570101973718701637937658, −5.32934022854546959518915104735, −5.10936939565739499317444558303, −3.79043597657823495495071445117, −3.03811386096925232362991129339, −2.22031333104271334851466946909, −1.62933292483041555808314994482, 0.75306985645082385276288917335, 1.66687674596639526043692889468, 2.510500732233599698710893083524, 3.45360747135344379628714183399, 4.213301283815464039067378251825, 4.91632797612785761831857783509, 5.9442912050817579498045605086, 6.337823756682697968153692054964, 7.11873538936344259267608136245, 8.115493110240123823621953939387, 9.24293909775926727205872737158, 9.819873518612201543938208110641, 10.6306554811960777421649654329, 11.31661615037356605923666212022, 11.95771327692741126804508556157, 13.08074619999531291894869097145, 13.62937381127562415367575041019, 13.70906053139856062138689175738, 14.6554918379458232961830273919, 15.57714740718263252049988980660, 16.38144529306515415747489706184, 16.8328655000721804440741438859, 17.53268941381008534945999213652, 18.700921541828338220738883625028, 19.26627803425286197909410893991

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