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Properties

Label 1-51e2-2601.970-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.862 - 0.505i$
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  + (−0.696 + 0.717i)2-s + (−0.0307 − 0.999i)4-s + (0.213 + 0.976i)5-s + (0.650 + 0.759i)7-s + (0.739 + 0.673i)8-s + (−0.850 − 0.526i)10-s + (0.881 + 0.473i)11-s + (−0.952 + 0.303i)13-s + (−0.998 − 0.0615i)14-s + (−0.998 + 0.0615i)16-s + (−0.273 + 0.961i)19-s + (0.969 − 0.243i)20-s + (−0.952 + 0.303i)22-s + (0.332 − 0.943i)23-s + (−0.908 + 0.417i)25-s + (0.445 − 0.895i)26-s + ⋯
L(s)  = 1  + (−0.696 + 0.717i)2-s + (−0.0307 − 0.999i)4-s + (0.213 + 0.976i)5-s + (0.650 + 0.759i)7-s + (0.739 + 0.673i)8-s + (−0.850 − 0.526i)10-s + (0.881 + 0.473i)11-s + (−0.952 + 0.303i)13-s + (−0.998 − 0.0615i)14-s + (−0.998 + 0.0615i)16-s + (−0.273 + 0.961i)19-s + (0.969 − 0.243i)20-s + (−0.952 + 0.303i)22-s + (0.332 − 0.943i)23-s + (−0.908 + 0.417i)25-s + (0.445 − 0.895i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (970, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.862 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2107012834 + 0.7760281731i\)
\(L(\frac12)\) \(\approx\) \(-0.2107012834 + 0.7760281731i\)
\(L(1)\) \(\approx\) \(0.5782839283 + 0.4980982195i\)
\(L(1)\) \(\approx\) \(0.5782839283 + 0.4980982195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.696 + 0.717i)T \)
5 \( 1 + (0.213 + 0.976i)T \)
7 \( 1 + (0.650 + 0.759i)T \)
11 \( 1 + (0.881 + 0.473i)T \)
13 \( 1 + (-0.952 + 0.303i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.332 - 0.943i)T \)
29 \( 1 + (0.881 + 0.473i)T \)
31 \( 1 + (-0.952 + 0.303i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (-0.908 - 0.417i)T \)
43 \( 1 + (0.552 - 0.833i)T \)
47 \( 1 + (0.332 + 0.943i)T \)
53 \( 1 + (-0.982 + 0.183i)T \)
59 \( 1 + (-0.779 + 0.626i)T \)
61 \( 1 + (-0.779 - 0.626i)T \)
67 \( 1 + (-0.696 - 0.717i)T \)
71 \( 1 + (-0.982 + 0.183i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (0.969 - 0.243i)T \)
83 \( 1 + (-0.908 + 0.417i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (0.650 + 0.759i)T \)
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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

−19.30059238818577536257339705839, −18.077786427081113373924343888609, −17.38313894265298534112730911919, −17.14581731270152039801193364990, −16.469788100265432830643605729322, −15.61561853174855549453013320303, −14.5738758226504634317492307859, −13.69731860279732826301376118363, −13.19224543126973418132970448578, −12.336596786125278302803457170956, −11.69026011810104368609418409884, −11.062173909115282872128992174826, −10.24186193852444828315068663612, −9.42930026927810384796109581661, −8.95144844309730867441869696230, −8.11974204909079741333183374842, −7.47056879335724477044857252634, −6.6671300528776171477896660686, −5.39045363181494383022007530429, −4.59664577516029157442074378127, −3.9768267408376031206769065763, −2.981948584685800512208765230207, −1.88737862929581094868246007955, −1.222593988512482436930428843222, −0.32657768328133570298942937012, 1.52571749797835480414178006093, 2.04596000427965781048615150661, 3.05383041235476880182153878237, 4.37578136621352759137855617914, 5.08630930011931282235584812593, 6.02400131801361159001613659092, 6.63093706900771661028606312930, 7.30447567428972868736278454608, 8.049567417863296813530631092294, 8.950717527821882530995588423010, 9.46594047863521904763632402182, 10.422579149754804976510437995330, 10.817391235347401439471915395052, 11.93297896222839731495207417309, 12.340511885650932719947052189607, 13.86494317794901057804429267257, 14.37332295189305651267298214681, 14.8345498984737606833567899560, 15.35925864307292871996477264423, 16.33117605700164184988534077537, 17.19740459404788362985705648644, 17.51944394764943950657364403372, 18.47220164463190004970035573461, 18.767105141367842245778492131613, 19.50606712749654282098790172020

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