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Properties

Label 1-712-712.35-r0-0-0
Degree $1$
Conductor $712$
Sign $0.908 - 0.418i$
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  + (−0.212 − 0.977i)3-s + (−0.755 − 0.654i)5-s + (−0.997 + 0.0713i)7-s + (−0.909 + 0.415i)9-s + (0.654 + 0.755i)11-s + (0.977 − 0.212i)13-s + (−0.479 + 0.877i)15-s + (−0.281 + 0.959i)17-s + (0.936 + 0.349i)19-s + (0.281 + 0.959i)21-s + (−0.349 + 0.936i)23-s + (0.142 + 0.989i)25-s + (0.599 + 0.800i)27-s + (−0.0713 − 0.997i)29-s + (−0.349 − 0.936i)31-s + ⋯
L(s)  = 1  + (−0.212 − 0.977i)3-s + (−0.755 − 0.654i)5-s + (−0.997 + 0.0713i)7-s + (−0.909 + 0.415i)9-s + (0.654 + 0.755i)11-s + (0.977 − 0.212i)13-s + (−0.479 + 0.877i)15-s + (−0.281 + 0.959i)17-s + (0.936 + 0.349i)19-s + (0.281 + 0.959i)21-s + (−0.349 + 0.936i)23-s + (0.142 + 0.989i)25-s + (0.599 + 0.800i)27-s + (−0.0713 − 0.997i)29-s + (−0.349 − 0.936i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.908 - 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9094914610 - 0.1996562381i\)
\(L(\frac12)\) \(\approx\) \(0.9094914610 - 0.1996562381i\)
\(L(1)\) \(\approx\) \(0.7916792020 - 0.2147014723i\)
\(L(1)\) \(\approx\) \(0.7916792020 - 0.2147014723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.212 - 0.977i)T \)
5 \( 1 + (-0.755 - 0.654i)T \)
7 \( 1 + (-0.997 + 0.0713i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.977 - 0.212i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.936 + 0.349i)T \)
23 \( 1 + (-0.349 + 0.936i)T \)
29 \( 1 + (-0.0713 - 0.997i)T \)
31 \( 1 + (-0.349 - 0.936i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.977 + 0.212i)T \)
43 \( 1 + (0.0713 - 0.997i)T \)
47 \( 1 + (0.540 + 0.841i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.212 - 0.977i)T \)
61 \( 1 + (0.800 - 0.599i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.755 - 0.654i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (-0.479 - 0.877i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−22.55444648340353233852627827490, −22.1015436187076787220075170272, −21.12055904989390635039017540500, −20.07948419492063816867492289990, −19.623008702559230812291171727237, −18.57697291941756473933494039153, −17.87062350250680118098141908617, −16.37729880448724620897319946524, −16.216693964503356751781470313369, −15.55708708267983725919065233008, −14.36984572007464772112083830706, −13.90279501483125052345503119810, −12.53808666347720936893708135927, −11.5498548158333183878400948228, −11.01829614023436466158269370090, −10.1765592296904252860212147339, −9.15304672315878464166896313088, −8.575915919700951243434303276625, −7.10818856139420020110945072718, −6.4334305037793260316285593690, −5.43853211218198551434914573732, −4.17273913446685143307551238986, −3.474125658763976816819394453236, −2.83794802469934976136006111960, −0.64784905248081952776598512423, 0.89187743601039210465111395678, 1.90733687099060828813704902811, 3.373129926964205565205201457954, 4.129631367113463856169822604876, 5.58124459703489357012347903600, 6.262746176851880504660101783, 7.272974449466912278550901548036, 8.004197206582669668158704383171, 8.93491351253933200296687512538, 9.82172116607193329575840837724, 11.16384661171581050728156498279, 11.879469695576066908440540804492, 12.63481192085732714395419543511, 13.18520787612226636861431870708, 14.09164912985716875097496571343, 15.39197255657383027842999562094, 15.9318829290343909061504013093, 17.00412637100684125172787286715, 17.51529891502754051072006419475, 18.74697094376072409490838534909, 19.18597209731833779269073324447, 20.0906318682023625478042400213, 20.503938026837514834965230116458, 22.02752595902460521809658420782, 22.76574282202874697126627148861

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