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Properties

Label 1-712-712.509-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.453 - 0.891i$
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  + (−0.841 − 0.540i)3-s + (0.654 − 0.755i)5-s + (−0.654 + 0.755i)7-s + (0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.841 − 0.540i)13-s + (−0.959 + 0.281i)15-s + (−0.959 − 0.281i)17-s + (−0.415 − 0.909i)19-s + (0.959 − 0.281i)21-s + (0.415 + 0.909i)23-s + (−0.142 − 0.989i)25-s + (0.142 − 0.989i)27-s + (0.654 − 0.755i)29-s + (0.415 − 0.909i)31-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)3-s + (0.654 − 0.755i)5-s + (−0.654 + 0.755i)7-s + (0.415 + 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.841 − 0.540i)13-s + (−0.959 + 0.281i)15-s + (−0.959 − 0.281i)17-s + (−0.415 − 0.909i)19-s + (0.959 − 0.281i)21-s + (0.415 + 0.909i)23-s + (−0.142 − 0.989i)25-s + (0.142 − 0.989i)27-s + (0.654 − 0.755i)29-s + (0.415 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4026511395 - 0.6563246444i\)
\(L(\frac12)\) \(\approx\) \(0.4026511395 - 0.6563246444i\)
\(L(1)\) \(\approx\) \(0.7333425979 - 0.2520413597i\)
\(L(1)\) \(\approx\) \(0.7333425979 - 0.2520413597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 - T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
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

−22.63944260366736513156634250112, −22.132483047191000004268820786966, −21.508095882482042007212648014272, −20.61233373695127137593414840720, −19.45820766945404373491677402769, −18.84614414538140517444404396525, −17.71717481794205622615823701153, −17.113400843184367135594953126090, −16.503775897911047815128983727388, −15.634358782379574802696614738238, −14.47058724027392652137347882275, −14.028409176798478671009459958376, −12.81657776174902622146647383364, −12.02518295464309532913184667338, −10.77153867079146737429752673933, −10.59104298022368833336654899412, −9.59795532440898905493578292765, −8.80108710763362452048667670706, −7.09893761335252782915581360196, −6.56817076812273461195880325846, −5.895735062698895694797411863861, −4.6329810394194062512452806224, −3.7956658031569078957268345830, −2.74711955777996168152115099094, −1.24834443685828904005861578346, 0.433531243961541754216892198324, 1.851493635658714473643260862288, 2.60485462106964370681239115771, 4.44434666587692286335646166815, 5.12032249604097077460688274993, 6.06741079698334751647260833248, 6.73638946535127304782368168937, 7.765469987576025087291096949582, 9.08659388653466233577913843867, 9.56700592089952319734682542676, 10.63621631731893103753527709933, 11.80538126017606861901224768541, 12.35331373859148462805599678679, 13.07877525437962755829587856886, 13.73618797128816088871011326272, 15.18465592447824786827706496099, 15.79585890174511974857249294994, 16.88149184872660664735513527543, 17.54258074349391479177912148603, 17.87409738979768606871004750613, 19.27451105115401848515690112863, 19.6172814948404227279440990385, 20.78460270327955143569299218893, 21.82078139773642782666294796946, 22.28713616509423125444442289064

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