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Properties

Label 1-51e2-2601.2245-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.999 + 0.00241i$
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  + (0.650 − 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.881 + 0.473i)5-s + (−0.389 − 0.920i)7-s + (−0.850 − 0.526i)8-s + (0.932 − 0.361i)10-s + (−0.779 + 0.626i)11-s + (−0.0307 + 0.999i)13-s + (−0.952 − 0.303i)14-s + (−0.952 + 0.303i)16-s + (−0.982 + 0.183i)19-s + (0.332 − 0.943i)20-s + (−0.0307 + 0.999i)22-s + (0.992 + 0.122i)23-s + (0.552 + 0.833i)25-s + (0.739 + 0.673i)26-s + ⋯
L(s)  = 1  + (0.650 − 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.881 + 0.473i)5-s + (−0.389 − 0.920i)7-s + (−0.850 − 0.526i)8-s + (0.932 − 0.361i)10-s + (−0.779 + 0.626i)11-s + (−0.0307 + 0.999i)13-s + (−0.952 − 0.303i)14-s + (−0.952 + 0.303i)16-s + (−0.982 + 0.183i)19-s + (0.332 − 0.943i)20-s + (−0.0307 + 0.999i)22-s + (0.992 + 0.122i)23-s + (0.552 + 0.833i)25-s + (0.739 + 0.673i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.999 + 0.00241i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (2245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.999 + 0.00241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.992019208 + 0.002406042274i\)
\(L(\frac12)\) \(\approx\) \(1.992019208 + 0.002406042274i\)
\(L(1)\) \(\approx\) \(1.350353745 - 0.4262720384i\)
\(L(1)\) \(\approx\) \(1.350353745 - 0.4262720384i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.650 - 0.759i)T \)
5 \( 1 + (0.881 + 0.473i)T \)
7 \( 1 + (-0.389 - 0.920i)T \)
11 \( 1 + (-0.779 + 0.626i)T \)
13 \( 1 + (-0.0307 + 0.999i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (0.992 + 0.122i)T \)
29 \( 1 + (-0.779 + 0.626i)T \)
31 \( 1 + (-0.0307 + 0.999i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (0.552 - 0.833i)T \)
43 \( 1 + (0.213 + 0.976i)T \)
47 \( 1 + (0.992 - 0.122i)T \)
53 \( 1 + (-0.602 + 0.798i)T \)
59 \( 1 + (0.969 - 0.243i)T \)
61 \( 1 + (0.969 + 0.243i)T \)
67 \( 1 + (0.650 + 0.759i)T \)
71 \( 1 + (-0.602 + 0.798i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.332 - 0.943i)T \)
83 \( 1 + (0.552 + 0.833i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (-0.389 - 0.920i)T \)
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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

−19.1779884125947437559125911683, −18.51058885533235871600353909844, −17.81311609033091113979892936864, −17.038687005004273934606511633165, −16.59612593490078811927515938805, −15.60909072416132000711165155944, −15.246390793373758244940125936506, −14.48259604623724373829213062865, −13.44906740159033511122294509515, −13.02433193752647471277647120988, −12.670132534304524092406040974, −11.64895931476198622286912804563, −10.76216917114431298210734052672, −9.78131354867948655949290763193, −9.00288352332682052438285487784, −8.38975584919371152646326179204, −7.72119174735395775737081260991, −6.53531614063394837386397522991, −5.98535453957006638785037726066, −5.37788306497753587547355544962, −4.82520508699886729436019037249, −3.63936666782617260367819233801, −2.706203426587267129993023272046, −2.229314157128531111848810988234, −0.50946729484065030125951350725, 1.092986634785089226054191508848, 2.01191258846792660482147871353, 2.6332400531980893643991350245, 3.610661632945887038106737105535, 4.34378660746271121670544045507, 5.17435132546507625045732986402, 5.95517801017533584148466631445, 6.89063186801386448939041528782, 7.219498357955306587031778341493, 8.78383267643421759731570427064, 9.49714978792457077270816544669, 10.13344989725759502155592353310, 10.82067688765390913571644292946, 11.178107909262996139342553811262, 12.58284135983624778824992049115, 12.82349096099384709834032339974, 13.620781023339112241743017887, 14.291346804206111585387014743365, 14.73430150567946074107325648153, 15.70696784416545941058362558751, 16.55523963968892534252056742410, 17.40609797361063033504315863669, 18.013229342324055122676923551023, 18.9660640850362106697094691659, 19.25728126092224952995979689853

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