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Properties

Label 1-712-712.229-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.769 - 0.638i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (−0.877 − 0.479i)3-s + (0.909 + 0.415i)5-s + (−0.936 + 0.349i)7-s + (0.540 + 0.841i)9-s + (0.415 + 0.909i)11-s + (−0.479 + 0.877i)13-s + (−0.599 − 0.800i)15-s + (−0.989 + 0.142i)17-s + (−0.212 + 0.977i)19-s + (0.989 + 0.142i)21-s + (−0.977 − 0.212i)23-s + (0.654 + 0.755i)25-s + (−0.0713 − 0.997i)27-s + (0.349 + 0.936i)29-s + (−0.977 + 0.212i)31-s + ⋯
L(s)  = 1  + (−0.877 − 0.479i)3-s + (0.909 + 0.415i)5-s + (−0.936 + 0.349i)7-s + (0.540 + 0.841i)9-s + (0.415 + 0.909i)11-s + (−0.479 + 0.877i)13-s + (−0.599 − 0.800i)15-s + (−0.989 + 0.142i)17-s + (−0.212 + 0.977i)19-s + (0.989 + 0.142i)21-s + (−0.977 − 0.212i)23-s + (0.654 + 0.755i)25-s + (−0.0713 − 0.997i)27-s + (0.349 + 0.936i)29-s + (−0.977 + 0.212i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07197268843 + 0.1993372809i\)
\(L(\frac12)\) \(\approx\) \(-0.07197268843 + 0.1993372809i\)
\(L(1)\) \(\approx\) \(0.6883183044 + 0.1486254152i\)
\(L(1)\) \(\approx\) \(0.6883183044 + 0.1486254152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.877 - 0.479i)T \)
5 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (-0.936 + 0.349i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.479 + 0.877i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (-0.212 + 0.977i)T \)
23 \( 1 + (-0.977 - 0.212i)T \)
29 \( 1 + (0.349 + 0.936i)T \)
31 \( 1 + (-0.977 + 0.212i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.479 - 0.877i)T \)
43 \( 1 + (0.349 - 0.936i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.877 - 0.479i)T \)
61 \( 1 + (0.997 - 0.0713i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (-0.599 + 0.800i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−21.98273294289937057953383705717, −21.40040667524023072202442264869, −20.22546530554323368023288884389, −19.67963529117035284662904498310, −18.42998279892123943888925173268, −17.52662098673907329055756617842, −17.12446662345171565093655665877, −16.17144608544187665417459834967, −15.68368053986883027592296038807, −14.47234251913802257487787787579, −13.32395748355035195316552965331, −12.95560624384654283233215217855, −11.84139708459694846011743096578, −10.923475458043138921524230338615, −10.09308228710775293189246937866, −9.4743069582430011386061018642, −8.60807937397794502201216592004, −7.07434582671735314787698441103, −6.23257646027849312708874039576, −5.646221918514255692898081550, −4.6302363338547126272472352201, −3.62428964621397318395705019670, −2.41898995065727520740056223767, −0.85583533605987849700788904102, −0.06592123510102569934565023373, 1.73735384485614728926813794068, 2.20892145368887899412757261780, 3.77730321869043224269643174092, 4.96251474506505841677110104950, 5.896842148570991630166568364986, 6.73729478751773007511014291846, 7.04193440501573859689297682420, 8.65459742156724326108733704894, 9.71818437750639170673159947145, 10.21551400620323891972398266408, 11.25408592534647911528404777717, 12.346546898794036858627196468767, 12.67679809626866231529139589524, 13.78104356337580751820637899723, 14.50643611561571806247112341930, 15.68468949025678358887598773657, 16.52809047873646779115799494613, 17.26422308928936647126325216461, 17.96350979755376564848612203573, 18.71379704994477094539297274239, 19.41724736045666575917917285733, 20.45567023116591856630078123919, 21.63644264575816888990909805854, 22.3148358675925486623289621593, 22.52566701396073544016582080026

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