L-function 1-2652-2652.2243-r1-0-0

• Label: 1-2652-2652.2243-r1-0-0
• Degree: $1$
• Conductor: $2652$
• Sign: $0.477 + 0.878i$
• Analytic cond.: $284.996$
• Root an. cond.: $284.996$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·5^-s + (0.866 + 0.5i)7^-s + (0.866 − 0.5i)11^-s + (0.866 + 0.5i)19^-s + (−0.5 − 0.866i)23^-s − 25^-s + (−0.5 − 0.866i)29^-s − i·31^-s + (−0.5 + 0.866i)35^-s + (0.866 − 0.5i)37^-s + (0.866 − 0.5i)41^-s + (−0.5 + 0.866i)43^-s − i·47^-s + (0.5 + 0.866i)49^-s − 53^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2652$    =    $2^{2} \cdot 3 \cdot 13 \cdot 17$
Sign:	$0.477 + 0.878i$
Analytic conductor:	$284.996$
Root analytic conductor:	$284.996$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2652} (2243, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ 0.477 + 0.878i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.559593899 + 1.522122879i$
$L(1)$	$\approx$	$1.285611711 + 0.3055850345i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	13	$1$
	17	$1$
good	5	$1 + iT$
	7	$1 + (0.866 + 0.5i)T$
	11	$1 + (0.866 - 0.5i)T$
	19	$1 + (0.866 + 0.5i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (-0.5 - 0.866i)T$
	31	$1 - iT$
	37	$1 + (0.866 - 0.5i)T$
	41	$1 + (0.866 - 0.5i)T$

Imaginary part of the first few zeros on the critical line

−19.197676035498589661645729492501, −18.02403292145593140850948337931, −17.65164192773858909194171559149, −16.896298571484268698088319890807, −16.37569119869470183077211761184, −15.48851731482063135288714815494, −14.74596421665856695183908985071, −14.01270059944897349998719312735, −13.35108174227407984144220030457, −12.624248527002606571945254267383, −11.64666261538788512689305820767, −11.46735471316232200117021186971, −10.31848214946515084392268575835, −9.39075772678598210074249907131, −9.07776247685553183713028731881, −7.87730734113956890706396120117, −7.62093906011517717976447776318, −6.532396928217930037658876151249, −5.574608791293535824360564256522, −4.836393567809420545750987575659, −4.23883246205766367009476443744, −3.42997285979465851678967926584, −2.04213718706056401295939311569, −1.36461004975215361383084936232, −0.61968459784728297088319886859, 0.80027070495741818022496391527, 1.84278391977088998950865673133, 2.597537116617167139766277069656, 3.535957432723143539679306751911, 4.24883505839736865250867867073, 5.33436282891229954840336319452, 6.04775303503131141354258328911, 6.73178275627681306581278006048, 7.66338493681011374052778533452, 8.24862851116774322878600579974, 9.14635161597833470266857890394, 9.912785559893336847917036357823, 10.7429345355325504542942127025, 11.470487973568385646498771457901, 11.835906052957041878306688899963, 12.792407351750736901015964075268, 13.918166942778748545826213946622, 14.335584932816147357403423465166, 14.81460827917059442132351190061, 15.689904119263256802465981281954, 16.407174643238154335730040618336, 17.289575465611871639903510774579, 18.020991801509162052662706651993, 18.45249639225064515605526829908, 19.18529115431603589321270035380

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