Properties

Label 1-1441-1441.135-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.418 - 0.908i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.399 − 0.916i)2-s + (−0.998 − 0.0483i)3-s + (−0.681 − 0.732i)4-s + (0.836 − 0.548i)5-s + (−0.443 + 0.896i)6-s + (0.885 + 0.464i)7-s + (−0.943 + 0.331i)8-s + (0.995 + 0.0965i)9-s + (−0.168 − 0.985i)10-s + (0.644 + 0.764i)12-s + (0.885 + 0.464i)13-s + (0.779 − 0.626i)14-s + (−0.861 + 0.506i)15-s + (−0.0724 + 0.997i)16-s + (0.644 − 0.764i)17-s + (0.485 − 0.873i)18-s + ⋯
L(s)  = 1  + (0.399 − 0.916i)2-s + (−0.998 − 0.0483i)3-s + (−0.681 − 0.732i)4-s + (0.836 − 0.548i)5-s + (−0.443 + 0.896i)6-s + (0.885 + 0.464i)7-s + (−0.943 + 0.331i)8-s + (0.995 + 0.0965i)9-s + (−0.168 − 0.985i)10-s + (0.644 + 0.764i)12-s + (0.885 + 0.464i)13-s + (0.779 − 0.626i)14-s + (−0.861 + 0.506i)15-s + (−0.0724 + 0.997i)16-s + (0.644 − 0.764i)17-s + (0.485 − 0.873i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.418 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541975559 - 0.9867131029i\)
\(L(\frac12)\) \(\approx\) \(1.541975559 - 0.9867131029i\)
\(L(1)\) \(\approx\) \(1.087907765 - 0.5898541203i\)
\(L(1)\) \(\approx\) \(1.087907765 - 0.5898541203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.399 - 0.916i)T \)
3 \( 1 + (-0.998 - 0.0483i)T \)
5 \( 1 + (0.836 - 0.548i)T \)
7 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (0.644 - 0.764i)T \)
19 \( 1 + (0.926 - 0.377i)T \)
23 \( 1 + (-0.607 + 0.794i)T \)
29 \( 1 + (-0.748 + 0.663i)T \)
31 \( 1 + (0.0241 + 0.999i)T \)
37 \( 1 + (0.120 + 0.992i)T \)
41 \( 1 + (-0.527 + 0.849i)T \)
43 \( 1 + (0.836 - 0.548i)T \)
47 \( 1 + (0.995 + 0.0965i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.644 - 0.764i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.836 + 0.548i)T \)
71 \( 1 + (0.958 + 0.285i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.779 + 0.626i)T \)
83 \( 1 + (-0.681 + 0.732i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.989 + 0.144i)T \)
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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

−21.01976540753614670247720006720, −20.62498414405711586953577380979, −18.84597816386701322723607376090, −18.23231602859054200488795689274, −17.75397102565084755292798890289, −17.03918854399669025680635805518, −16.519694378482973108691862653759, −15.57345683809956527752537289512, −14.81006772396603498882993168570, −14.072933435016247714428231990065, −13.40474680324963398492960700045, −12.6109684551574809773875707776, −11.70652560563628856771527112969, −10.83292679584380914275160584492, −10.181318212608776621162493227528, −9.26747276985016223219805499939, −8.0347714094483130837232219070, −7.472968467735641026384154504, −6.51262093910669473160108862662, −5.72089178827827700635635178386, −5.45953143386770110790825058884, −4.23381047932484608513001000582, −3.59880655199464332515662596282, −2.05954670996286731685984733657, −0.86359204572540200615876068816, 1.19499645539897993081416694015, 1.43680810113805191980017378885, 2.63553506781734286313207086968, 3.89931816085512841487173804908, 4.86923200107556400959600448114, 5.407695136059363150734846491352, 5.913306001998646467840324463573, 7.07434251845615631274490934555, 8.37044011985447493246645782609, 9.27636787619814622272585770743, 9.865194020502960318516071711731, 10.81861947065545653228339236662, 11.484226767525116486475834552096, 12.0487257333546169059399585629, 12.76281122550116139292120021822, 13.7862085343995380395778906348, 14.02014334377413255836629052379, 15.31995335032060481969828011998, 16.10187332525159676175850332227, 17.020312670350074995111645946407, 17.85474064988014008187658844019, 18.271048108554177634620513296906, 18.841117720238709049671708635080, 20.2179565067540494157151195415, 20.68164577804046895823378134126

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