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Properties

Label 1-61-61.24-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.348 + 0.937i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + (0.587 + 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.951 + 0.309i)6-s + (−0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)10-s + i·11-s + (0.309 + 0.951i)12-s + 13-s − 14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.309i)17-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.951 + 0.309i)6-s + (−0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)10-s + i·11-s + (0.309 + 0.951i)12-s + 13-s − 14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.348 + 0.937i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.348 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338183594 + 1.924887076i\)
\(L(\frac12)\) \(\approx\) \(1.338183594 + 1.924887076i\)
\(L(1)\) \(\approx\) \(1.353972449 + 0.9322737156i\)
\(L(1)\) \(\approx\) \(1.353972449 + 0.9322737156i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + iT \)
13 \( 1 + T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + iT \)
31 \( 1 + (0.587 - 0.809i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.587 - 0.809i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−32.24707774734274823726808381430, −30.95014936617194186846182738692, −29.85505469732386909764234079170, −28.647828795082899622916955899786, −27.568118631258420452268737700629, −26.629191083932337079591385880407, −25.10877558043454035738837708518, −23.88284867816259311320979117877, −22.80448132521864208574031665804, −21.378980087627349804346920733, −20.58673527498668563134077735475, −19.78114724218851045606291450183, −18.76724706566584320563160806526, −16.4996637373539664816808411141, −15.69419175260461754299768885065, −13.92254798497392190170475375356, −13.45744133410103146383389426777, −11.91776680433701964768911387539, −10.445793629680005730858117493577, −9.38992967784235095071040248260, −8.08536798523911227733911957825, −5.685779473610725071092225427798, −4.116935753128568424795582140691, −3.29207599617430988123054809318, −1.06191087791363180198707646871, 2.62889655673528986470610526529, 3.77606091870044792197265947554, 5.99899207748416571634677703853, 7.06921896376327045618910039321, 8.16535252055528924560771631405, 9.59194639393865433214939530567, 11.8641579704300607204781152766, 12.90268856313529726630270791005, 14.164621788944932131941029624597, 15.075312976433463780250671454273, 15.95806949922015016943619681199, 17.9792660434558167366135078288, 18.61088335734234175243274093793, 20.09789621234350956554879834612, 21.541907166228847992774205576856, 22.73019190396107588314580974669, 23.59861839977087063452379877422, 24.96986964457012986495355896364, 25.80612112096674007449720352569, 26.35067294821649362430986185860, 28.05481000217349166243659500932, 29.895425734274843671372279773, 30.66526872582716308605679692550, 31.4097498566201178017663646230, 32.47126325448455980998521585598

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