Properties

Label 2-23-23.20-c2-0-2
Degree $2$
Conductor $23$
Sign $0.181 + 0.983i$
Analytic cond. $0.626704$
Root an. cond. $0.791646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.387 − 2.69i)2-s + (−0.141 + 0.309i)3-s + (−3.28 + 0.963i)4-s + (−0.353 − 0.306i)5-s + (0.890 + 0.261i)6-s + (4.93 + 7.67i)7-s + (−0.655 − 1.43i)8-s + (5.81 + 6.71i)9-s + (−0.689 + 1.07i)10-s + (−9.31 − 1.33i)11-s + (0.165 − 1.15i)12-s + (−15.9 − 10.2i)13-s + (18.7 − 16.2i)14-s + (0.145 − 0.0662i)15-s + (−15.1 + 9.72i)16-s + (2.33 − 7.94i)17-s + ⋯
L(s)  = 1  + (−0.193 − 1.34i)2-s + (−0.0471 + 0.103i)3-s + (−0.820 + 0.240i)4-s + (−0.0707 − 0.0613i)5-s + (0.148 + 0.0435i)6-s + (0.704 + 1.09i)7-s + (−0.0819 − 0.179i)8-s + (0.646 + 0.745i)9-s + (−0.0689 + 0.107i)10-s + (−0.846 − 0.121i)11-s + (0.0138 − 0.0961i)12-s + (−1.23 − 0.790i)13-s + (1.34 − 1.16i)14-s + (0.00967 − 0.00441i)15-s + (−0.945 + 0.607i)16-s + (0.137 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(0.626704\)
Root analytic conductor: \(0.791646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :1),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.643101 - 0.535286i\)
\(L(\frac12)\) \(\approx\) \(0.643101 - 0.535286i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-22.6 + 3.86i)T \)
good2 \( 1 + (0.387 + 2.69i)T + (-3.83 + 1.12i)T^{2} \)
3 \( 1 + (0.141 - 0.309i)T + (-5.89 - 6.80i)T^{2} \)
5 \( 1 + (0.353 + 0.306i)T + (3.55 + 24.7i)T^{2} \)
7 \( 1 + (-4.93 - 7.67i)T + (-20.3 + 44.5i)T^{2} \)
11 \( 1 + (9.31 + 1.33i)T + (116. + 34.0i)T^{2} \)
13 \( 1 + (15.9 + 10.2i)T + (70.2 + 153. i)T^{2} \)
17 \( 1 + (-2.33 + 7.94i)T + (-243. - 156. i)T^{2} \)
19 \( 1 + (-4.88 - 16.6i)T + (-303. + 195. i)T^{2} \)
29 \( 1 + (20.1 + 5.92i)T + (707. + 454. i)T^{2} \)
31 \( 1 + (-8.08 - 17.6i)T + (-629. + 726. i)T^{2} \)
37 \( 1 + (-48.8 + 42.2i)T + (194. - 1.35e3i)T^{2} \)
41 \( 1 + (12.8 - 14.8i)T + (-239. - 1.66e3i)T^{2} \)
43 \( 1 + (-13.8 - 6.30i)T + (1.21e3 + 1.39e3i)T^{2} \)
47 \( 1 + 66.5T + 2.20e3T^{2} \)
53 \( 1 + (13.4 + 20.9i)T + (-1.16e3 + 2.55e3i)T^{2} \)
59 \( 1 + (-43.0 - 27.6i)T + (1.44e3 + 3.16e3i)T^{2} \)
61 \( 1 + (54.7 - 25.0i)T + (2.43e3 - 2.81e3i)T^{2} \)
67 \( 1 + (-70.5 + 10.1i)T + (4.30e3 - 1.26e3i)T^{2} \)
71 \( 1 + (2.48 + 17.2i)T + (-4.83e3 + 1.42e3i)T^{2} \)
73 \( 1 + (-56.1 + 16.4i)T + (4.48e3 - 2.88e3i)T^{2} \)
79 \( 1 + (-44.6 + 69.4i)T + (-2.59e3 - 5.67e3i)T^{2} \)
83 \( 1 + (92.3 - 80.0i)T + (980. - 6.81e3i)T^{2} \)
89 \( 1 + (-160. - 73.1i)T + (5.18e3 + 5.98e3i)T^{2} \)
97 \( 1 + (-71.9 - 62.3i)T + (1.33e3 + 9.31e3i)T^{2} \)
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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

−17.99124623481004580761859422620, −16.12679648616573831937168596836, −14.87186684583257723639473124152, −12.99961271975182368307786449562, −12.06964067291214312890552633579, −10.80416378333172268190846677005, −9.690960749525414380415292002049, −7.930121717949516694673985465341, −5.06575209771210617266417028383, −2.46337879441383850237669113282, 4.82911154333015480853863855709, 6.91230134780693029825178938496, 7.70945396385944363386082227036, 9.551451944077321098121688561279, 11.34221806467525497762727322872, 13.20284553517118458673225145736, 14.60547812154018305956532777783, 15.39494727964788716155104475884, 16.86480643279953784428470182855, 17.46639229909268724610369514180

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