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Properties

Label 2-525-35.17-c1-0-4
Degree $2$
Conductor $525$
Sign $-0.902 - 0.431i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (−0.355 + 1.32i)2-s + (−0.965 + 0.258i)3-s + (0.100 + 0.0578i)4-s − 1.37i·6-s + (1.91 + 1.82i)7-s + (−2.05 + 2.05i)8-s + (0.866 − 0.499i)9-s + (0.557 − 0.966i)11-s + (−0.111 − 0.0299i)12-s + (0.707 + 0.707i)13-s + (−3.10 + 1.88i)14-s + (−1.87 − 3.25i)16-s + (0.941 + 3.51i)17-s + (0.355 + 1.32i)18-s + (3.00 + 5.20i)19-s + ⋯
L(s)  = 1  + (−0.251 + 0.937i)2-s + (−0.557 + 0.149i)3-s + (0.0501 + 0.0289i)4-s − 0.560i·6-s + (0.722 + 0.691i)7-s + (−0.726 + 0.726i)8-s + (0.288 − 0.166i)9-s + (0.168 − 0.291i)11-s + (−0.0322 − 0.00864i)12-s + (0.196 + 0.196i)13-s + (−0.829 + 0.503i)14-s + (−0.469 − 0.813i)16-s + (0.228 + 0.852i)17-s + (0.0837 + 0.312i)18-s + (0.689 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244847 + 1.08052i\)
\(L(\frac12)\) \(\approx\) \(0.244847 + 1.08052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-1.91 - 1.82i)T \)
good2 \( 1 + (0.355 - 1.32i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-0.557 + 0.966i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \)
17 \( 1 + (-0.941 - 3.51i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.00 - 5.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.91 + 1.31i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.23iT - 29T^{2} \)
31 \( 1 + (4.40 + 2.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.05 - 7.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.76iT - 41T^{2} \)
43 \( 1 + (-6.33 + 6.33i)T - 43iT^{2} \)
47 \( 1 + (6.87 + 1.84i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.42 + 12.7i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.0470 - 0.0814i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.88 + 1.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.04T + 71T^{2} \)
73 \( 1 + (-7.99 + 2.14i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.76 + 2.17i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.03 - 5.03i)T + 83iT^{2} \)
89 \( 1 + (-1.52 - 2.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.8 + 11.8i)T - 97iT^{2} \)
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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

−11.45147935586922111229292962303, −10.32828133473714950879004928791, −9.301609497732155363751322896055, −8.205258834470265545455717711512, −7.85486660650126724480576669555, −6.46766213099856733983925242963, −5.91534964067921113200338497294, −5.04295856117001796802448062144, −3.59749726681917317166870277644, −1.89089292638754940179254794510, 0.77080279304903451777275462125, 2.01366878867438943216518032368, 3.44782699806803589992560710375, 4.68295646729364749301393379382, 5.76069436246141014493989285380, 6.98001558920796894001164593996, 7.60238642563903049790808170546, 9.062134772168316066418644937877, 9.792761506554046507977274446978, 10.86734273158768505939512108594

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