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Properties

Label 1-51e2-2601.511-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.176 - 0.984i$
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  + (0.332 + 0.943i)2-s + (−0.779 + 0.626i)4-s + (−0.0307 − 0.999i)5-s + (0.992 + 0.122i)7-s + (−0.850 − 0.526i)8-s + (0.932 − 0.361i)10-s + (−0.153 − 0.988i)11-s + (0.881 − 0.473i)13-s + (0.213 + 0.976i)14-s + (0.213 − 0.976i)16-s + (−0.982 + 0.183i)19-s + (0.650 + 0.759i)20-s + (0.881 − 0.473i)22-s + (−0.389 − 0.920i)23-s + (−0.998 + 0.0615i)25-s + (0.739 + 0.673i)26-s + ⋯
L(s)  = 1  + (0.332 + 0.943i)2-s + (−0.779 + 0.626i)4-s + (−0.0307 − 0.999i)5-s + (0.992 + 0.122i)7-s + (−0.850 − 0.526i)8-s + (0.932 − 0.361i)10-s + (−0.153 − 0.988i)11-s + (0.881 − 0.473i)13-s + (0.213 + 0.976i)14-s + (0.213 − 0.976i)16-s + (−0.982 + 0.183i)19-s + (0.650 + 0.759i)20-s + (0.881 − 0.473i)22-s + (−0.389 − 0.920i)23-s + (−0.998 + 0.0615i)25-s + (0.739 + 0.673i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.176 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9353594271 - 0.7829364797i\)
\(L(\frac12)\) \(\approx\) \(0.9353594271 - 0.7829364797i\)
\(L(1)\) \(\approx\) \(1.091982396 + 0.1243492400i\)
\(L(1)\) \(\approx\) \(1.091982396 + 0.1243492400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.332 + 0.943i)T \)
5 \( 1 + (-0.0307 - 0.999i)T \)
7 \( 1 + (0.992 + 0.122i)T \)
11 \( 1 + (-0.153 - 0.988i)T \)
13 \( 1 + (0.881 - 0.473i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (-0.389 - 0.920i)T \)
29 \( 1 + (-0.153 - 0.988i)T \)
31 \( 1 + (0.881 - 0.473i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (-0.998 - 0.0615i)T \)
43 \( 1 + (-0.952 - 0.303i)T \)
47 \( 1 + (-0.389 + 0.920i)T \)
53 \( 1 + (-0.602 + 0.798i)T \)
59 \( 1 + (-0.696 - 0.717i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (0.332 - 0.943i)T \)
71 \( 1 + (-0.602 + 0.798i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.650 + 0.759i)T \)
83 \( 1 + (-0.998 + 0.0615i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.992 + 0.122i)T \)
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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

−19.66302690722090977804685429481, −18.737806456345493792639678839700, −18.26048078136669841051104184432, −17.71729707009597505650155056171, −16.99594113755046360141671919808, −15.622812016512180897190213346534, −15.01855509139284312658184539114, −14.5283281823293553093495834520, −13.71024617919357731283376883213, −13.240136513909774375653742707804, −12.085891538588480004496658917931, −11.628385630811235839799197106921, −10.89792924157202861178019935678, −10.368169115905939264941539062647, −9.67120480836855080060018268405, −8.64955840018121417332478839334, −7.99515379075194578204865501102, −6.91395134103353864991820577306, −6.26808109306870162396829078559, −5.161168913355148577159038338894, −4.53054469628657968022180299403, −3.70651039505360467405136489013, −2.9474932275578830664348856780, −1.83833316064516545958916640835, −1.56120888027011768379627889851, 0.33699662736288066191258799297, 1.3885215360936119656612126749, 2.62853968201046498034854062328, 3.84141812016554631911946323331, 4.38235562121575542073035587913, 5.19112385827704989103468959206, 5.88886043490328632368527991639, 6.44023638617258590110972648751, 7.79646489006082712586203486273, 8.28406909891294890859631033461, 8.57302559707973382886166949394, 9.50089948800071455256204174242, 10.60850759607235146333526981413, 11.42907209383489505271214865370, 12.2451645381723612028709932161, 12.92242839905585717095473553145, 13.6643650298276739161204158655, 14.12125803596675979800515817453, 15.14612598899311345963896103112, 15.60559062692069083900189330455, 16.40790968473736146973016561048, 16.992471475721854247752434177229, 17.534481054552124537023218997923, 18.419730951200633074290532600192, 18.92422828372586384080823577984

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