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Properties

Label 1-59-59.37-r1-0-0
Degree $1$
Conductor $59$
Sign $0.321 - 0.947i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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  + (0.947 − 0.319i)2-s + (−0.856 + 0.515i)3-s + (0.796 − 0.605i)4-s + (−0.370 − 0.928i)5-s + (−0.647 + 0.762i)6-s + (0.907 + 0.419i)7-s + (0.561 − 0.827i)8-s + (0.468 − 0.883i)9-s + (−0.647 − 0.762i)10-s + (−0.267 − 0.963i)11-s + (−0.370 + 0.928i)12-s + (−0.468 − 0.883i)13-s + (0.994 + 0.108i)14-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (0.907 − 0.419i)17-s + ⋯
L(s)  = 1  + (0.947 − 0.319i)2-s + (−0.856 + 0.515i)3-s + (0.796 − 0.605i)4-s + (−0.370 − 0.928i)5-s + (−0.647 + 0.762i)6-s + (0.907 + 0.419i)7-s + (0.561 − 0.827i)8-s + (0.468 − 0.883i)9-s + (−0.647 − 0.762i)10-s + (−0.267 − 0.963i)11-s + (−0.370 + 0.928i)12-s + (−0.468 − 0.883i)13-s + (0.994 + 0.108i)14-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (0.907 − 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.321 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.321 - 0.947i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ 0.321 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.717364656 - 1.231090237i\)
\(L(\frac12)\) \(\approx\) \(1.717364656 - 1.231090237i\)
\(L(1)\) \(\approx\) \(1.420989726 - 0.5060404648i\)
\(L(1)\) \(\approx\) \(1.420989726 - 0.5060404648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.947 - 0.319i)T \)
3 \( 1 + (-0.856 + 0.515i)T \)
5 \( 1 + (-0.370 - 0.928i)T \)
7 \( 1 + (0.907 + 0.419i)T \)
11 \( 1 + (-0.267 - 0.963i)T \)
13 \( 1 + (-0.468 - 0.883i)T \)
17 \( 1 + (0.907 - 0.419i)T \)
19 \( 1 + (0.976 + 0.214i)T \)
23 \( 1 + (0.161 + 0.986i)T \)
29 \( 1 + (-0.947 - 0.319i)T \)
31 \( 1 + (-0.976 + 0.214i)T \)
37 \( 1 + (0.561 + 0.827i)T \)
41 \( 1 + (-0.161 + 0.986i)T \)
43 \( 1 + (-0.267 + 0.963i)T \)
47 \( 1 + (0.370 - 0.928i)T \)
53 \( 1 + (0.647 - 0.762i)T \)
61 \( 1 + (0.947 - 0.319i)T \)
67 \( 1 + (0.561 - 0.827i)T \)
71 \( 1 + (-0.370 + 0.928i)T \)
73 \( 1 + (0.994 + 0.108i)T \)
79 \( 1 + (-0.856 - 0.515i)T \)
83 \( 1 + (-0.0541 + 0.998i)T \)
89 \( 1 + (0.947 + 0.319i)T \)
97 \( 1 + (0.994 - 0.108i)T \)
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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

−33.02901796835365147517666446054, −31.15757248251931381517220940720, −30.57070493251878754479400419836, −29.722524116531431951012958917134, −28.47319077810102016959336709653, −26.988223894716024807036009531413, −25.76244991590709438150417176740, −24.30520739129286373274360425177, −23.581133585336532025331080104653, −22.68145260270771420335572620160, −21.72900045500953152728147367994, −20.33112892529613079287675389800, −18.707587700175747759207773780269, −17.52625131938336336837066353720, −16.414920219752701053720616725956, −14.91671582251464261862916586510, −14.045712796635805359014344939052, −12.48633861950980442121395682426, −11.53314697845070685174633720782, −10.54044557684325085761754801072, −7.5623318906298835370409542272, −7.07448997636338265075148844620, −5.46302155996582233360306168077, −4.17058920374182343622908073179, −2.07464613954734004306716050605, 1.04025007609459498230455061679, 3.51277895819485173095672154090, 5.148234065535748991709968338, 5.52544451447600972137283891724, 7.7639970482447614292008556076, 9.710021561307130824496131963320, 11.26453216648561273224101987013, 11.904846205348297740433299535102, 13.14256261458460078120661816336, 14.786521477600961060313068516166, 15.85939866595583623746115444833, 16.821883508574660815911224173552, 18.42657316037871135936994960216, 20.13710503611640739661810051138, 21.07474394904021421389476834189, 21.90202954628523318648485803206, 23.18539104437719257881470309405, 24.0831288131184895111669914316, 24.90901839788136936626165650594, 27.190453488173694348006375654373, 27.87694686190916491750011428460, 28.9726655269739651205368722749, 29.90355327439627024500214299762, 31.43792850081718976625469966885, 32.125874449040778051332854520660

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