L-function 1-3040-3040.1309-r0-0-0

• Label: 1-3040-3040.1309-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.388 + 0.921i$
• Analytic cond.: $14.1177$
• Root an. cond.: $14.1177$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.573 − 0.819i)3^-s + (−0.866 − 0.5i)7^-s + (−0.342 − 0.939i)9^-s + (−0.965 − 0.258i)11^-s + (0.819 − 0.573i)13^-s + (−0.939 − 0.342i)17^-s + (−0.906 + 0.422i)21^-s + (0.642 − 0.766i)23^-s + (−0.965 − 0.258i)27^-s + (−0.906 − 0.422i)29^-s + (−0.5 + 0.866i)31^-s + (−0.766 + 0.642i)33^-s + (0.707 − 0.707i)37^-s − i·39^-s + (−0.984 + 0.173i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.388 + 0.921i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1309, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.388 + 0.921i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.1763990965 - 0.2657253200i$
$L(1)$	$\approx$	$0.7869919601 - 0.4174354884i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (0.573 - 0.819i)T$
	7	$1 + (-0.866 - 0.5i)T$
	11	$1 + (-0.965 - 0.258i)T$
	13	$1 + (0.819 - 0.573i)T$
	17	$1 + (-0.939 - 0.342i)T$
	23	$1 + (0.642 - 0.766i)T$
	29	$1 + (-0.906 - 0.422i)T$
	31	$1 + (-0.5 + 0.866i)T$
	37	$1 + (0.707 - 0.707i)T$

Imaginary part of the first few zeros on the critical line

−19.6380035381841322892318249940, −18.67427152182832607533868125064, −18.5019178306509919341302631914, −17.22872188301644225318922950633, −16.60337624062490897312213761205, −15.8017850004053233473667765739, −15.4499829230849990256512047365, −14.86764512048270784723303802787, −13.83848376416953035764069866528, −13.16486157763015675835410852455, −12.82347895855086023669737365300, −11.44897286338143188048039928084, −11.07402965872642565058940775701, −10.09170384900624538846950961987, −9.57485842782776687972321760078, −8.84337487750276903299870363370, −8.30332687073973085388665418268, −7.3188231028512103392808065706, −6.48068687390986083569835632046, −5.56748078180102959549783065945, −4.93717413590123041719144377461, −3.90657040609376391260210069468, −3.37940785760119398675230523866, −2.47431809291129631813172135536, −1.77324451619120258500749565421, 0.09109748157654037680746633044, 1.03472901413825345249774886256, 2.14378520063172413773354327673, 2.99177139043782995233028884687, 3.47228964348102975771657632685, 4.53908923951431350855158349050, 5.631134247300943120879429158749, 6.380571486409220582442058284697, 6.99341399603123539109405952238, 7.760498059338813660386387614065, 8.45273518101760963141178434067, 9.14560154023144385117342594062, 9.94965264446805034018646698644, 10.83637396936925052175874571162, 11.39022410600973710388305768447, 12.61207037393460487491071727911, 13.07861032516909432657452151632, 13.35188506449053686405549973471, 14.24278578005406840070901514834, 15.02942762273721085742556978987, 15.81103880647789847841099358309, 16.34829438917703151155906436730, 17.29728926656136835719320102626, 18.137520967154006965643836216012, 18.463512815212295892679727411840

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