Properties

Label 1-4011-4011.215-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.311 + 0.950i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.815 − 0.578i)2-s + (0.329 + 0.944i)4-s + (0.904 − 0.426i)5-s + (0.277 − 0.960i)8-s + (−0.984 − 0.175i)10-s + (0.191 − 0.981i)11-s + (−0.371 − 0.928i)13-s + (−0.782 + 0.622i)16-s + (0.00551 + 0.999i)17-s + (−0.984 + 0.175i)19-s + (0.701 + 0.712i)20-s + (−0.724 + 0.689i)22-s + (0.899 + 0.436i)23-s + (0.635 − 0.771i)25-s + (−0.234 + 0.972i)26-s + ⋯
L(s)  = 1  + (−0.815 − 0.578i)2-s + (0.329 + 0.944i)4-s + (0.904 − 0.426i)5-s + (0.277 − 0.960i)8-s + (−0.984 − 0.175i)10-s + (0.191 − 0.981i)11-s + (−0.371 − 0.928i)13-s + (−0.782 + 0.622i)16-s + (0.00551 + 0.999i)17-s + (−0.984 + 0.175i)19-s + (0.701 + 0.712i)20-s + (−0.724 + 0.689i)22-s + (0.899 + 0.436i)23-s + (0.635 − 0.771i)25-s + (−0.234 + 0.972i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.311 + 0.950i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1094852594 + 0.1511214577i\)
\(L(\frac12)\) \(\approx\) \(0.1094852594 + 0.1511214577i\)
\(L(1)\) \(\approx\) \(0.6680605898 - 0.2029275335i\)
\(L(1)\) \(\approx\) \(0.6680605898 - 0.2029275335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.815 - 0.578i)T \)
5 \( 1 + (0.904 - 0.426i)T \)
11 \( 1 + (0.191 - 0.981i)T \)
13 \( 1 + (-0.371 - 0.928i)T \)
17 \( 1 + (0.00551 + 0.999i)T \)
19 \( 1 + (-0.984 + 0.175i)T \)
23 \( 1 + (0.899 + 0.436i)T \)
29 \( 1 + (-0.894 - 0.446i)T \)
31 \( 1 + (-0.0275 + 0.999i)T \)
37 \( 1 + (0.0275 + 0.999i)T \)
41 \( 1 + (-0.677 + 0.735i)T \)
43 \( 1 + (-0.934 + 0.355i)T \)
47 \( 1 + (0.224 + 0.974i)T \)
53 \( 1 + (-0.874 - 0.485i)T \)
59 \( 1 + (-0.942 - 0.335i)T \)
61 \( 1 + (0.949 + 0.314i)T \)
67 \( 1 + (-0.739 - 0.673i)T \)
71 \( 1 + (-0.980 + 0.197i)T \)
73 \( 1 + (-0.996 + 0.0880i)T \)
79 \( 1 + (-0.834 - 0.551i)T \)
83 \( 1 + (0.828 + 0.560i)T \)
89 \( 1 + (-0.889 + 0.456i)T \)
97 \( 1 + (-0.991 + 0.131i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.2682931067639769767501082253, −17.50351711571190464937608054701, −16.96980601608482856994269343092, −16.54739072571136777563348497733, −15.5195931046839880535380258289, −14.81428054033424907369578489467, −14.49776289592603916568500282992, −13.65594818352383093772359288993, −12.95165508487739296633417008553, −11.9282834541517987881112722450, −11.19568481734805670338483319815, −10.46809112448176608576702988969, −9.85373294316805816667745915260, −9.14531113877571035578599876499, −8.85596800485550453383521175221, −7.55683372271910685764723436187, −6.99951736156511076776693402447, −6.6150816221981836309773030834, −5.66084384365091963165772069017, −4.986728604699639046201099797233, −4.19407858952779927387111375526, −2.76799288964956847092367918784, −2.08029028808790066270488767051, −1.51948859983589756301142196897, −0.064224098022180461135226534809, 1.237245834650804410701553280428, 1.6638637568244541220635504711, 2.80230062140498881661586251231, 3.29543837634357409164106863205, 4.35019201763738949974613706131, 5.27405460269677980856177049640, 6.1336890239241611617092234719, 6.69623449564315105924123882558, 7.85007763732793946497828445473, 8.39228236513350310576868690403, 8.96970908766710546239227245845, 9.73895660380874514363794633179, 10.38581506943154004974600067017, 10.907717851682496149292225130543, 11.69877837575812660248429172668, 12.64464877985987711867182502865, 13.02122447785520167481405108533, 13.61354724252875003596814329103, 14.682409199831826778182058935958, 15.32536393837941704671709178753, 16.33565761641425666046673077389, 16.87409051820584254743398367068, 17.35414617641392377097834579405, 17.88288236059270405596200804709, 18.82568936408742248023774902175

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