L-function 1-812-812.515-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	7	$1$
	29	$1$
good	3	$1 + (0.365 - 0.930i)T$
	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	$1 + (0.733 + 0.680i)T$

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