Properties

Label 2-23-23.4-c3-0-1
Degree $2$
Conductor $23$
Sign $0.437 - 0.899i$
Analytic cond. $1.35704$
Root an. cond. $1.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 3.52i)2-s + (3.27 − 0.961i)3-s + (−4.59 + 5.30i)4-s + (−10.7 − 6.91i)5-s + (8.66 + 9.99i)6-s + (−0.249 − 1.73i)7-s + (3.63 + 1.06i)8-s + (−12.9 + 8.29i)9-s + (7.05 − 49.0i)10-s + (21.0 − 46.1i)11-s + (−9.95 + 21.8i)12-s + (−2.51 + 17.5i)13-s + (5.70 − 3.66i)14-s + (−41.8 − 12.3i)15-s + (10.0 + 70.1i)16-s + (8.23 + 9.50i)17-s + ⋯
L(s)  = 1  + (0.569 + 1.24i)2-s + (0.630 − 0.185i)3-s + (−0.574 + 0.663i)4-s + (−0.962 − 0.618i)5-s + (0.589 + 0.680i)6-s + (−0.0134 − 0.0935i)7-s + (0.160 + 0.0471i)8-s + (−0.478 + 0.307i)9-s + (0.223 − 1.55i)10-s + (0.577 − 1.26i)11-s + (−0.239 + 0.524i)12-s + (−0.0537 + 0.373i)13-s + (0.108 − 0.0700i)14-s + (−0.721 − 0.211i)15-s + (0.157 + 1.09i)16-s + (0.117 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28722 + 0.805458i\)
\(L(\frac12)\) \(\approx\) \(1.28722 + 0.805458i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-92.7 + 59.7i)T \)
good2 \( 1 + (-1.61 - 3.52i)T + (-5.23 + 6.04i)T^{2} \)
3 \( 1 + (-3.27 + 0.961i)T + (22.7 - 14.5i)T^{2} \)
5 \( 1 + (10.7 + 6.91i)T + (51.9 + 113. i)T^{2} \)
7 \( 1 + (0.249 + 1.73i)T + (-329. + 96.6i)T^{2} \)
11 \( 1 + (-21.0 + 46.1i)T + (-871. - 1.00e3i)T^{2} \)
13 \( 1 + (2.51 - 17.5i)T + (-2.10e3 - 618. i)T^{2} \)
17 \( 1 + (-8.23 - 9.50i)T + (-699. + 4.86e3i)T^{2} \)
19 \( 1 + (100. - 116. i)T + (-976. - 6.78e3i)T^{2} \)
29 \( 1 + (119. + 137. i)T + (-3.47e3 + 2.41e4i)T^{2} \)
31 \( 1 + (-84.3 - 24.7i)T + (2.50e4 + 1.61e4i)T^{2} \)
37 \( 1 + (-253. + 162. i)T + (2.10e4 - 4.60e4i)T^{2} \)
41 \( 1 + (-392. - 251. i)T + (2.86e4 + 6.26e4i)T^{2} \)
43 \( 1 + (48.0 - 14.0i)T + (6.68e4 - 4.29e4i)T^{2} \)
47 \( 1 + 189.T + 1.03e5T^{2} \)
53 \( 1 + (58.1 + 404. i)T + (-1.42e5 + 4.19e4i)T^{2} \)
59 \( 1 + (10.4 - 72.3i)T + (-1.97e5 - 5.78e4i)T^{2} \)
61 \( 1 + (182. + 53.5i)T + (1.90e5 + 1.22e5i)T^{2} \)
67 \( 1 + (1.94 + 4.26i)T + (-1.96e5 + 2.27e5i)T^{2} \)
71 \( 1 + (360. + 789. i)T + (-2.34e5 + 2.70e5i)T^{2} \)
73 \( 1 + (362. - 418. i)T + (-5.53e4 - 3.85e5i)T^{2} \)
79 \( 1 + (86.2 - 600. i)T + (-4.73e5 - 1.38e5i)T^{2} \)
83 \( 1 + (692. - 444. i)T + (2.37e5 - 5.20e5i)T^{2} \)
89 \( 1 + (-366. + 107. i)T + (5.93e5 - 3.81e5i)T^{2} \)
97 \( 1 + (998. + 641. i)T + (3.79e5 + 8.30e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.77967903009933308585362478031, −16.43726286435492990153454816413, −14.94816370434728472969936101579, −14.16455429741555509460579606867, −12.92353344191114760789409697967, −11.23833655703235401588263369449, −8.613762706736051925921279846332, −7.86164622424312633424226292475, −6.04649487020936814777192836481, −4.11000408089982977934835770753, 2.82680246725671525481186668944, 4.25508930641164860564019359118, 7.31619951031364695471968776473, 9.263485565163298822881395631142, 10.87377438476698588205138480877, 11.85085448222298543983386984449, 13.05452041156845562809429790386, 14.62602809528107813958522826691, 15.31898977027611458772940153500, 17.35998919491227120731349031291

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