Properties

Label 2-575-23.6-c1-0-17
Degree $2$
Conductor $575$
Sign $-0.575 - 0.818i$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
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  + (−0.683 + 1.49i)2-s + (3.06 + 0.900i)3-s + (−0.465 − 0.536i)4-s + (−3.44 + 3.97i)6-s + (0.0908 − 0.632i)7-s + (−2.03 + 0.598i)8-s + (6.07 + 3.90i)9-s + (1.69 + 3.71i)11-s + (−0.943 − 2.06i)12-s + (−0.459 − 3.19i)13-s + (0.884 + 0.568i)14-s + (0.699 − 4.86i)16-s + (0.836 − 0.965i)17-s + (−9.99 + 6.42i)18-s + (−0.307 − 0.354i)19-s + ⋯
L(s)  = 1  + (−0.483 + 1.05i)2-s + (1.77 + 0.519i)3-s + (−0.232 − 0.268i)4-s + (−1.40 + 1.62i)6-s + (0.0343 − 0.238i)7-s + (−0.720 + 0.211i)8-s + (2.02 + 1.30i)9-s + (0.512 + 1.12i)11-s + (−0.272 − 0.596i)12-s + (−0.127 − 0.885i)13-s + (0.236 + 0.151i)14-s + (0.174 − 1.21i)16-s + (0.202 − 0.234i)17-s + (−2.35 + 1.51i)18-s + (−0.0705 − 0.0814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.575 - 0.818i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ -0.575 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966041 + 1.85973i\)
\(L(\frac12)\) \(\approx\) \(0.966041 + 1.85973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 + (3.57 - 3.19i)T \)
good2 \( 1 + (0.683 - 1.49i)T + (-1.30 - 1.51i)T^{2} \)
3 \( 1 + (-3.06 - 0.900i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (-0.0908 + 0.632i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-1.69 - 3.71i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.459 + 3.19i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-0.836 + 0.965i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.307 + 0.354i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-2.61 + 3.02i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (4.96 - 1.45i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (8.04 + 5.16i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-7.36 + 4.73i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-0.704 - 0.206i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 0.589T + 47T^{2} \)
53 \( 1 + (-0.955 + 6.64i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.734 - 5.11i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (2.44 - 0.717i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-0.998 + 2.18i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (3.46 - 7.58i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.483 - 0.557i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (2.33 + 16.2i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (8.77 + 5.64i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (5.75 + 1.68i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (5.42 - 3.48i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.49473283692937458173997052073, −9.737174881664538100205954944304, −9.084988612027900183671227306368, −8.361487403834108051337342854922, −7.44837545405309704341518505123, −7.19431736762068676823360801008, −5.61748460107944530136831016344, −4.30811650505242728335743853787, −3.30695333500388012387826739140, −2.13627778696505586544853927142, 1.31416441329295302150136987246, 2.29638545501269687769448355892, 3.22194247609655573653996122647, 4.05943629470357825044593186766, 6.10100670478205124158118063291, 7.03236473627952011129771116602, 8.263717592734247032218711484920, 8.784547778074609588306951539653, 9.390444504195577464174466772603, 10.24179570434319574429733590983

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