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Properties

Label 1-712-712.547-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.584 - 0.811i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.0713 + 0.997i)3-s + (0.281 − 0.959i)5-s + (−0.479 − 0.877i)7-s + (−0.989 − 0.142i)9-s + (0.959 − 0.281i)11-s + (−0.997 − 0.0713i)13-s + (0.936 + 0.349i)15-s + (−0.909 − 0.415i)17-s + (0.599 + 0.800i)19-s + (0.909 − 0.415i)21-s + (−0.800 + 0.599i)23-s + (−0.841 − 0.540i)25-s + (0.212 − 0.977i)27-s + (0.877 − 0.479i)29-s + (−0.800 − 0.599i)31-s + ⋯
L(s)  = 1  + (−0.0713 + 0.997i)3-s + (0.281 − 0.959i)5-s + (−0.479 − 0.877i)7-s + (−0.989 − 0.142i)9-s + (0.959 − 0.281i)11-s + (−0.997 − 0.0713i)13-s + (0.936 + 0.349i)15-s + (−0.909 − 0.415i)17-s + (0.599 + 0.800i)19-s + (0.909 − 0.415i)21-s + (−0.800 + 0.599i)23-s + (−0.841 − 0.540i)25-s + (0.212 − 0.977i)27-s + (0.877 − 0.479i)29-s + (−0.800 − 0.599i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2529169217 - 0.4941145980i\)
\(L(\frac12)\) \(\approx\) \(0.2529169217 - 0.4941145980i\)
\(L(1)\) \(\approx\) \(0.7964110816 - 0.07016731063i\)
\(L(1)\) \(\approx\) \(0.7964110816 - 0.07016731063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.0713 + 0.997i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (-0.479 - 0.877i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (-0.997 - 0.0713i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (0.599 + 0.800i)T \)
23 \( 1 + (-0.800 + 0.599i)T \)
29 \( 1 + (0.877 - 0.479i)T \)
31 \( 1 + (-0.800 - 0.599i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.997 + 0.0713i)T \)
43 \( 1 + (-0.877 - 0.479i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.0713 + 0.997i)T \)
61 \( 1 + (-0.977 - 0.212i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.281 - 0.959i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.989 + 0.142i)T \)
83 \( 1 + (0.936 - 0.349i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−22.69063816752186692443274600718, −22.12232846370624551238516900797, −21.716657841585527456494646120719, −19.962556709391152201535278535176, −19.63310440524841032989879287569, −18.80209366462865358446447051568, −17.9146442671912563193942495778, −17.59535502083282553733486788768, −16.43751304922115017098352984452, −15.290756057532013501045310448051, −14.54631453497677699983562459989, −13.885722732344301104234066655808, −12.853634508872778962928728464829, −12.089473075299365943252833323, −11.463191523209764295453732095202, −10.35797360372891965029947034210, −9.32583932768854396377044950698, −8.5632288645351379357590754528, −7.26189582533759700603509777304, −6.70588015531282934481840166995, −6.0511011713438023270013493654, −4.9107208380160343850211573283, −3.34649970948619467471999988225, −2.47935033751133074995348205336, −1.72787355524366351709069721819, 0.24821036239488965625099201680, 1.74480979629968630891901094010, 3.275425750331965291291201691619, 4.11571665995656201639148138481, 4.86404231847202641986024852029, 5.81728118842189408640041517582, 6.84499872751493203813837040383, 8.08193344865655356645036235652, 9.02674978268108665417418803488, 9.784826523407045230430772706640, 10.25008106202810337081584148687, 11.60905809896918972393806844646, 12.10929462984257378323988622869, 13.47380068513554910441188162250, 13.94067627011629446892311510375, 15.00369504630108352394866811233, 15.94763787557858271872077285203, 16.71270631424229425504107227573, 17.0270541942026959498792663454, 17.96185618371379591857615536172, 19.563389463972093004991207922536, 19.97256297495604699573228349517, 20.57099805644349710878337390048, 21.58794778141916380451957200616, 22.2260458834895202746720510338

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