Properties

Label 2-23-23.2-c3-0-0
Degree $2$
Conductor $23$
Sign $0.661 - 0.749i$
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  + (−3.24 − 2.08i)2-s + (1.21 + 8.45i)3-s + (2.84 + 6.23i)4-s + (12.7 + 3.75i)5-s + (13.6 − 29.9i)6-s + (−11.3 + 13.0i)7-s + (−0.631 + 4.38i)8-s + (−44.1 + 12.9i)9-s + (−33.6 − 38.8i)10-s + (45.2 − 29.0i)11-s + (−49.2 + 31.6i)12-s + (−27.2 − 31.4i)13-s + (64.0 − 18.8i)14-s + (−16.2 + 112. i)15-s + (47.0 − 54.3i)16-s + (14.8 − 32.4i)17-s + ⋯
L(s)  = 1  + (−1.14 − 0.736i)2-s + (0.234 + 1.62i)3-s + (0.355 + 0.778i)4-s + (1.14 + 0.336i)5-s + (0.930 − 2.03i)6-s + (−0.612 + 0.707i)7-s + (−0.0278 + 0.193i)8-s + (−1.63 + 0.480i)9-s + (−1.06 − 1.22i)10-s + (1.24 − 0.796i)11-s + (−1.18 + 0.761i)12-s + (−0.581 − 0.671i)13-s + (1.22 − 0.359i)14-s + (−0.279 + 1.94i)15-s + (0.735 − 0.848i)16-s + (0.211 − 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.697359 + 0.314700i\)
\(L(\frac12)\) \(\approx\) \(0.697359 + 0.314700i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-89.7 - 64.1i)T \)
good2 \( 1 + (3.24 + 2.08i)T + (3.32 + 7.27i)T^{2} \)
3 \( 1 + (-1.21 - 8.45i)T + (-25.9 + 7.60i)T^{2} \)
5 \( 1 + (-12.7 - 3.75i)T + (105. + 67.5i)T^{2} \)
7 \( 1 + (11.3 - 13.0i)T + (-48.8 - 339. i)T^{2} \)
11 \( 1 + (-45.2 + 29.0i)T + (552. - 1.21e3i)T^{2} \)
13 \( 1 + (27.2 + 31.4i)T + (-312. + 2.17e3i)T^{2} \)
17 \( 1 + (-14.8 + 32.4i)T + (-3.21e3 - 3.71e3i)T^{2} \)
19 \( 1 + (-8.29 - 18.1i)T + (-4.49e3 + 5.18e3i)T^{2} \)
29 \( 1 + (-77.8 + 170. i)T + (-1.59e4 - 1.84e4i)T^{2} \)
31 \( 1 + (0.302 - 2.10i)T + (-2.85e4 - 8.39e3i)T^{2} \)
37 \( 1 + (-99.0 + 29.0i)T + (4.26e4 - 2.73e4i)T^{2} \)
41 \( 1 + (457. + 134. i)T + (5.79e4 + 3.72e4i)T^{2} \)
43 \( 1 + (-1.21 - 8.42i)T + (-7.62e4 + 2.23e4i)T^{2} \)
47 \( 1 - 239.T + 1.03e5T^{2} \)
53 \( 1 + (-49.4 + 57.1i)T + (-2.11e4 - 1.47e5i)T^{2} \)
59 \( 1 + (-136. - 157. i)T + (-2.92e4 + 2.03e5i)T^{2} \)
61 \( 1 + (45.8 - 318. i)T + (-2.17e5 - 6.39e4i)T^{2} \)
67 \( 1 + (583. + 375. i)T + (1.24e5 + 2.73e5i)T^{2} \)
71 \( 1 + (-76.1 - 48.9i)T + (1.48e5 + 3.25e5i)T^{2} \)
73 \( 1 + (-44.1 - 96.5i)T + (-2.54e5 + 2.93e5i)T^{2} \)
79 \( 1 + (373. + 430. i)T + (-7.01e4 + 4.88e5i)T^{2} \)
83 \( 1 + (713. - 209. i)T + (4.81e5 - 3.09e5i)T^{2} \)
89 \( 1 + (79.7 + 554. i)T + (-6.76e5 + 1.98e5i)T^{2} \)
97 \( 1 + (538. + 158. i)T + (7.67e5 + 4.93e5i)T^{2} \)
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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

−17.45956818367491137832093911564, −16.65393784185072304419236931789, −15.18878926475722880345474034739, −14.00293191550847154878357230164, −11.68127190538685697764141531124, −10.32196200577914456358675281799, −9.600605217855692398170463931035, −8.862218792156276613401793248941, −5.66392178025128411109011058720, −2.95982300969810993228438341518, 1.39647407296357926090513752906, 6.52871288584274773499948194040, 7.10072357285910281103413289011, 8.795912708993995572419219098463, 9.869534417933488543010898976463, 12.35552077491752686357899851958, 13.35817050474939430536951234871, 14.55255527580054103212783734833, 16.84589229856499232754070347766, 17.18756106088494599731096401051

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