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Properties

Label 1-51e2-2601.409-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.968 - 0.249i$
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  + (0.552 + 0.833i)2-s + (−0.389 + 0.920i)4-s + (0.332 + 0.943i)5-s + (0.213 + 0.976i)7-s + (−0.982 + 0.183i)8-s + (−0.602 + 0.798i)10-s + (0.992 − 0.122i)11-s + (0.650 + 0.759i)13-s + (−0.696 + 0.717i)14-s + (−0.696 − 0.717i)16-s + (0.445 + 0.895i)19-s + (−0.998 − 0.0615i)20-s + (0.650 + 0.759i)22-s + (−0.952 − 0.303i)23-s + (−0.779 + 0.626i)25-s + (−0.273 + 0.961i)26-s + ⋯
L(s)  = 1  + (0.552 + 0.833i)2-s + (−0.389 + 0.920i)4-s + (0.332 + 0.943i)5-s + (0.213 + 0.976i)7-s + (−0.982 + 0.183i)8-s + (−0.602 + 0.798i)10-s + (0.992 − 0.122i)11-s + (0.650 + 0.759i)13-s + (−0.696 + 0.717i)14-s + (−0.696 − 0.717i)16-s + (0.445 + 0.895i)19-s + (−0.998 − 0.0615i)20-s + (0.650 + 0.759i)22-s + (−0.952 − 0.303i)23-s + (−0.779 + 0.626i)25-s + (−0.273 + 0.961i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.968 - 0.249i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.968 - 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3084984993 + 2.431153267i\)
\(L(\frac12)\) \(\approx\) \(-0.3084984993 + 2.431153267i\)
\(L(1)\) \(\approx\) \(0.9087800859 + 1.237814093i\)
\(L(1)\) \(\approx\) \(0.9087800859 + 1.237814093i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.552 + 0.833i)T \)
5 \( 1 + (0.332 + 0.943i)T \)
7 \( 1 + (0.213 + 0.976i)T \)
11 \( 1 + (0.992 - 0.122i)T \)
13 \( 1 + (0.650 + 0.759i)T \)
19 \( 1 + (0.445 + 0.895i)T \)
23 \( 1 + (-0.952 - 0.303i)T \)
29 \( 1 + (0.992 - 0.122i)T \)
31 \( 1 + (0.650 + 0.759i)T \)
37 \( 1 + (-0.850 + 0.526i)T \)
41 \( 1 + (-0.779 - 0.626i)T \)
43 \( 1 + (0.969 + 0.243i)T \)
47 \( 1 + (-0.952 + 0.303i)T \)
53 \( 1 + (0.739 - 0.673i)T \)
59 \( 1 + (0.816 - 0.577i)T \)
61 \( 1 + (0.816 + 0.577i)T \)
67 \( 1 + (0.552 - 0.833i)T \)
71 \( 1 + (0.739 - 0.673i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.998 - 0.0615i)T \)
83 \( 1 + (-0.779 + 0.626i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (0.213 + 0.976i)T \)
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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

−19.47020659906022705773873963337, −18.18273835065101345217809420389, −17.61938041843309769927197222089, −17.0609590042298340274035395950, −16.032468351025170166226224636089, −15.45045845190262264095239986016, −14.33651192976896872397066262155, −13.84943994520780009209706365159, −13.26878249766174675143298089041, −12.6226531737215311785692968762, −11.74064588941184903519634353864, −11.290453025646734161036094727256, −10.17053847527657451959386010707, −9.89159984584323027714792501214, −8.8724891766489232277902418228, −8.29890371834472035316020105378, −7.11829209133111619404236834945, −6.211946098714638151637749490742, −5.47269814749200445227253811793, −4.61215147593123662595857805380, −4.06323299578617418967317529493, −3.29044030150251651681844865404, −2.13110397424958832026180249427, −1.21365102979081219848878814289, −0.69415229496474943649881633982, 1.55903666775800659409313979981, 2.480330730372897285042127137592, 3.41174183308271728748713045719, 4.021526786633604975994573660951, 5.08938632359434309784580298593, 5.90561581611682893649029847973, 6.47183471096876339763805715439, 6.9510134849431613659834070311, 8.14788507038796839625624631471, 8.6076227383029151151612808664, 9.49076302186980878514801427856, 10.26338055266341247336695579890, 11.55128206833237026392020171802, 11.79139264201077743380724252571, 12.608887275184263552297194066833, 13.71621592016217018507427575914, 14.282189338092885209111315227653, 14.49640384885420393072155520482, 15.60398404673810498687165173596, 15.930467629950048962513855366905, 16.87435439196596290584307321853, 17.6422878354971805416152879529, 18.22246262826164443968790147243, 18.810122610891877317107133328262, 19.5652892106688266333826120365

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