$L(\chi,s)$, where $\chi$ is the Dirichlet character with label 1048.523

• Degree: $1$
• Conductor: $1048$
• Sign: $1$
• Motivic weight: $0$
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(χ,s)  = 1

+ 3^-s − 5^-s − 7^-s + 9^-s + 11^-s − 13^-s − 15^-s − 17^-s − 19^-s − 21^-s + 23^-s + 25^-s + 27^-s + 29^-s + 31^-s + 33^-s + 35^-s + 37^-s − 39^-s + 41^-s + 43^-s − 45^-s + 47^-s + 49^-s − 51^-s − 53^-s − 55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1048\)    =    \(2^{3} \cdot 131\)
Sign:	$1$
Motivic weight:	\(0\)
Character:	$\chi_{1048} (523, \cdot )$
Sato-Tate group:	$\mu(2)$
Primitive:	yes
Self-dual:	yes
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1048,\ (0:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\)	\(\approx\)	\(1.584221912\)
\(L(\chi,1)\)	\(\approx\)	\(1.183859034\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.56718153107034037495030481434, −20.52368137868376890796363248692, −19.665799159285760822903372773771, −19.425821237939540728397202277026, −18.923410400316654056026151842910, −17.60127961710355694346731329502, −16.721479416003325235951365179, −15.86276451950599134392655731391, −15.21209697194462697533507014942, −14.61204931535224935017667927903, −13.69044827075253557429412273804, −12.71184959189750184204693100485, −12.30016859659436169856441318583, −11.16727150803664807865363283781, −10.21306790244152565779316634171, −9.23351424705048732396454081212, −8.79493319372769948409627095293, −7.75860514252762772599658295166, −6.95302990726421847670956160160, −6.35530651971128876955491550170, −4.52351020293829141410291409938, −4.17123136250834920402394989016, −3.05062975092852984149397841624, −2.42575377988817026551361968070, −0.85251316093032440677723859792, 0.85251316093032440677723859792, 2.42575377988817026551361968070, 3.05062975092852984149397841624, 4.17123136250834920402394989016, 4.52351020293829141410291409938, 6.35530651971128876955491550170, 6.95302990726421847670956160160, 7.75860514252762772599658295166, 8.79493319372769948409627095293, 9.23351424705048732396454081212, 10.21306790244152565779316634171, 11.16727150803664807865363283781, 12.30016859659436169856441318583, 12.71184959189750184204693100485, 13.69044827075253557429412273804, 14.61204931535224935017667927903, 15.21209697194462697533507014942, 15.86276451950599134392655731391, 16.721479416003325235951365179, 17.60127961710355694346731329502, 18.923410400316654056026151842910, 19.425821237939540728397202277026, 19.665799159285760822903372773771, 20.52368137868376890796363248692, 21.56718153107034037495030481434

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