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Properties

Label 1-812-812.515-r1-0-0
Degree $1$
Conductor $812$
Sign $0.868 + 0.495i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (0.365 − 0.930i)3-s + (0.0747 − 0.997i)5-s + (−0.733 − 0.680i)9-s + (−0.733 + 0.680i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 + 0.866i)17-s + (0.365 + 0.930i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.900 + 0.433i)27-s + (0.826 + 0.563i)31-s + (0.365 + 0.930i)33-s + (0.733 + 0.680i)37-s + (−0.988 − 0.149i)39-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)3-s + (0.0747 − 0.997i)5-s + (−0.733 − 0.680i)9-s + (−0.733 + 0.680i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 + 0.866i)17-s + (0.365 + 0.930i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.900 + 0.433i)27-s + (0.826 + 0.563i)31-s + (0.365 + 0.930i)33-s + (0.733 + 0.680i)37-s + (−0.988 − 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.868 + 0.495i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157116793 + 0.3066771027i\)
\(L(\frac12)\) \(\approx\) \(1.157116793 + 0.3066771027i\)
\(L(1)\) \(\approx\) \(0.9618155649 - 0.3387357033i\)
\(L(1)\) \(\approx\) \(0.9618155649 - 0.3387357033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.365 + 0.930i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.826 + 0.563i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.988 - 0.149i)T \)
67 \( 1 + (-0.955 + 0.294i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (-0.733 - 0.680i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−21.83986450336793566571265071537, −21.34281928725954118947070983827, −20.46659995011333042675411131165, −19.57177616798320493839340911873, −18.76299250716330825337713051930, −18.123532848096043559237107755911, −16.96148989309037261054270004903, −16.157640146151526090328022125805, −15.5421573385822174080854668934, −14.62648966161123877390662565364, −13.98219524523299449431595170486, −13.372584468879923150198429723137, −11.71754734427192945182957451268, −11.28588560059807040478278294252, −10.255427989378642418591442124477, −9.74386893748589616661887396430, −8.73809266367126292353427247163, −7.80334426052189856592216487677, −6.861585322745611437276133781516, −5.7989708514131534660788510386, −4.85982096989502093032043383284, −3.85579130642493413910565102861, −2.891854362469155586215469581556, −2.281033773287808471785618752090, −0.269133663218478554476002590574, 1.010824976621437964928428016554, 1.82109214591164914004312918502, 2.934957320943303076902507826467, 4.060712958360472685867587740169, 5.345665919296863168960412938124, 5.89483277045147913326220334045, 7.198069986508898805243303126869, 8.08100560510246256870804552350, 8.39202900054123252549923173746, 9.71954774151181446586627119898, 10.33298479156680456623487155579, 11.93931069948053760596633754059, 12.28618407446166880280667512994, 13.12666208138409189867909948868, 13.69526133153391532398898275564, 14.82001900096597238485008886830, 15.522731844593202020513289437539, 16.60524698345379115765813942432, 17.427063271508101418310722067805, 18.02553615724017754305255742283, 18.90724995407710376293630561038, 19.84364487334867736951785783388, 20.36687078465946943449183209049, 21.00292541345298733893641736105, 22.108332847259386421738051724866

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