Properties

Label 2-23-23.6-c3-0-3
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

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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: $\chi_{23} (6, \cdot )$
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 \( 1 + (998. - 641. i)T + (3.79e5 - 8.30e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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