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Properties

Label 2-575-115.68-c1-0-2
Degree $2$
Conductor $575$
Sign $0.991 - 0.130i$
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  + (−1.97 + 1.97i)2-s + (−2.17 − 2.17i)3-s − 5.78i·4-s + 8.57·6-s + (7.47 + 7.47i)8-s + 6.43i·9-s + (−12.5 + 12.5i)12-s + (−0.654 − 0.654i)13-s − 17.9·16-s + (−12.6 − 12.6i)18-s + (3.39 + 3.39i)23-s − 32.4i·24-s + 2.58·26-s + (7.45 − 7.45i)27-s + 9.46i·29-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)2-s + (−1.25 − 1.25i)3-s − 2.89i·4-s + 3.49·6-s + (2.64 + 2.64i)8-s + 2.14i·9-s + (−3.63 + 3.63i)12-s + (−0.181 − 0.181i)13-s − 4.48·16-s + (−2.99 − 2.99i)18-s + (0.707 + 0.707i)23-s − 6.63i·24-s + 0.506·26-s + (1.43 − 1.43i)27-s + 1.75i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389117 + 0.0255519i\)
\(L(\frac12)\) \(\approx\) \(0.389117 + 0.0255519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.39 - 3.39i)T \)
good2 \( 1 + (1.97 - 1.97i)T - 2iT^{2} \)
3 \( 1 + (2.17 + 2.17i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (0.654 + 0.654i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 9.46iT - 29T^{2} \)
31 \( 1 - 1.62T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 0.430T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-9.02 + 9.02i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 + (-3.42 - 3.42i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−10.72520235191878602741979926211, −9.746210395786829689500565166899, −8.673080151407925640190896013720, −7.84592827273689259175561573496, −7.00694225165463172719678635598, −6.66451445540939738987887971199, −5.53864201979521679565929434666, −5.11386628073825913665624641547, −1.83477688484246294138434183372, −0.68826945757560675993448651162, 0.76655954134843580770223459369, 2.65634989926505105510862985081, 3.95069111615526584046905081399, 4.64151568187333601089299043687, 6.18691387709171907098345784815, 7.39692092983806080227472826128, 8.563264018673586155246345975439, 9.456109001146451051595194184784, 9.899811027022991106828758610044, 10.81434099841896625763246929635

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