L-function 1-2667-2667.86-r0-0-0

• Label: 1-2667-2667.86-r0-0-0
• Degree: $1$
• Conductor: $2667$
• Sign: $-0.821 - 0.569i$
• Analytic cond.: $12.3854$
• Root an. cond.: $12.3854$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.733 − 0.680i)2^-s + (0.0747 − 0.997i)4^-s + (0.826 − 0.563i)5^-s + (−0.623 − 0.781i)8^-s + (0.222 − 0.974i)10^-s + (0.998 + 0.0498i)11^-s + (−0.411 − 0.911i)13^-s + (−0.988 − 0.149i)16^-s + (0.969 − 0.246i)17^-s + 19^-s + (−0.5 − 0.866i)20^-s + (0.766 − 0.642i)22^-s + (−0.998 + 0.0498i)23^-s + (0.365 − 0.930i)25^-s + (−0.921 − 0.388i)26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2667$    =    $3 \cdot 7 \cdot 127$
Sign:	$-0.821 - 0.569i$
Analytic conductor:	$12.3854$
Root analytic conductor:	$12.3854$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2667} (86, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2667,\ (0:\ ),\ -0.821 - 0.569i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9246161292 - 2.957793747i$
$L(1)$	$\approx$	$1.406776431 - 1.199587140i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	127	$1$
good	2	$1 + (0.733 - 0.680i)T$
	5	$1 + (0.826 - 0.563i)T$
	11	$1 + (0.998 + 0.0498i)T$
	13	$1 + (-0.411 - 0.911i)T$
	17	$1 + (0.969 - 0.246i)T$
	19	$1 + T$
	23	$1 + (-0.998 + 0.0498i)T$
	29	$1 + (-0.0249 - 0.999i)T$
	31	$1 + (-0.969 - 0.246i)T$

Imaginary part of the first few zeros on the critical line

−19.72477725146585379344017033828, −18.63021847038276296065871570741, −18.12024470880323980864948382225, −17.30519545982427915091737984466, −16.62955283605117036735141526888, −16.25166511136130014648058353652, −15.10228915202315802647830164514, −14.37561278205479633673158046749, −14.217492463579677801807372895617, −13.43846936655846986711613490082, −12.53344130030032795878832485760, −11.83730944001569246234163417604, −11.25889520849437700419049190500, −10.07700904387764559809124247266, −9.46986846304527993306348581123, −8.68067224099412715949683939931, −7.69964184947154311685905836819, −6.884539785656305694089881668695, −6.49064651007472891308856645879, −5.55346130977974735559615918500, −5.03236051764585674472379096797, −3.81965722959584688503815813566, −3.38368483904237958904736699118, −2.26897132679529377419872210273, −1.4777870249508115692490199383, 0.74506806651008358376975718515, 1.489135955996207791888066360760, 2.372884502785875916314541057507, 3.24511099710097606804746763800, 4.07640615527687946311281372329, 4.938163476452066324259211312289, 5.69987444715572607574599393031, 6.08641078477551658989333697440, 7.215687855169849399305628074511, 8.14717828926954839673804779156, 9.29607540794661121002057174280, 9.73289995387089879486777503570, 10.25200631000736607851111256751, 11.362000965168697352166231217310, 11.96349511900907085204760183879, 12.61506083057229046778194171057, 13.25826831349852102823163465906, 14.06773924304322018212493425474, 14.44920960900267828295856876480, 15.30494613511837303663226110807, 16.227838120258989920403235955624, 16.857554508131284276485188760892, 17.79747411943375266627682689016, 18.28983616190881443883002588411, 19.24307456054857700924864086159

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