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Properties

Label 2-23e2-23.18-c1-0-15
Degree $2$
Conductor $529$
Sign $-0.415 - 0.909i$
Analytic cond. $4.22408$
Root an. cond. $2.05525$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.41 + 0.708i)2-s + (−1.04 + 1.20i)3-s + (3.63 + 2.33i)4-s + (−0.210 + 1.46i)5-s + (−3.37 + 2.16i)6-s + (−0.510 − 1.11i)7-s + (3.81 + 4.40i)8-s + (0.0651 + 0.453i)9-s + (−1.54 + 3.38i)10-s + (−3.40 + 0.999i)11-s + (−6.60 + 1.93i)12-s + (−0.0566 + 0.123i)13-s + (−0.439 − 3.05i)14-s + (−1.54 − 1.78i)15-s + (2.49 + 5.45i)16-s + (5.07 − 3.26i)17-s + ⋯
L(s)  = 1  + (1.70 + 0.500i)2-s + (−0.602 + 0.695i)3-s + (1.81 + 1.16i)4-s + (−0.0941 + 0.654i)5-s + (−1.37 + 0.884i)6-s + (−0.192 − 0.422i)7-s + (1.34 + 1.55i)8-s + (0.0217 + 0.151i)9-s + (−0.488 + 1.06i)10-s + (−1.02 + 0.301i)11-s + (−1.90 + 0.559i)12-s + (−0.0157 + 0.0343i)13-s + (−0.117 − 0.816i)14-s + (−0.398 − 0.460i)15-s + (0.623 + 1.36i)16-s + (1.23 − 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.415 - 0.909i$
Analytic conductor: \(4.22408\)
Root analytic conductor: \(2.05525\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 529,\ (\ :1/2),\ -0.415 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59313 + 2.48012i\)
\(L(\frac12)\) \(\approx\) \(1.59313 + 2.48012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-2.41 - 0.708i)T + (1.68 + 1.08i)T^{2} \)
3 \( 1 + (1.04 - 1.20i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (0.210 - 1.46i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.510 + 1.11i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (3.40 - 0.999i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (0.0566 - 0.123i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-5.07 + 3.26i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-2.70 - 1.74i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-3.86 + 2.48i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (1.88 + 2.17i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.554 - 3.85i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.145 - 1.01i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.07 - 2.39i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 0.00935T + 47T^{2} \)
53 \( 1 + (-3.90 - 8.55i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-1.93 + 4.24i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (8.87 + 10.2i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-9.33 - 2.74i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-0.668 - 0.196i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (4.46 + 2.86i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-4.45 + 9.76i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (0.249 + 1.73i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (5.76 - 6.65i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-2.56 + 17.8i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.32886113090527487971291344362, −10.46977380972878015390970731511, −9.819049103839840963043677727037, −7.83123255756943278060270049625, −7.30483917562953291540798885039, −6.22293281319718604629425058657, −5.30476502736160220163992074953, −4.74839081186234671094817254646, −3.60044813456785580776990054560, −2.70287455757809827391491726148, 1.19508596579069580197758046367, 2.71506516095939586296332333117, 3.77771769731580653641414544371, 5.16207924640185074400106752440, 5.54996247170632643950152558650, 6.47705215962963259347284116086, 7.49682632968288666544633937880, 8.742191065096200512031632594021, 10.13271012576475264382834041171, 10.97309066409183987946966573446

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