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Properties

Label 2-525-35.33-c1-0-7
Degree $2$
Conductor $525$
Sign $-0.241 - 0.970i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.241 - 0.970i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28428 + 1.64376i\)
\(L(\frac12)\) \(\approx\) \(1.28428 + 1.64376i\)
\(L(\frac{3}{2})\) 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

−11.03392055650504494929223088874, −10.11891487581456648199471176540, −9.061921005663021213203436856911, −8.500583515077683963123975687669, −7.18305148238383221890271767693, −6.75922126718093704409342010711, −5.59656291975545473597090798139, −4.74271164118864129145151973694, −3.29747625726015966537808640303, −2.05402500150220898994585219600, 1.17990590071030079675424248613, 2.72511394373545322769058428886, 3.46624677905024925947437073309, 4.44361867537027814614225189901, 6.01944573405177134602893396313, 7.23153919306755225855151755807, 7.67979740508334826282484184842, 9.177177515048826529976704364260, 9.811874308242568663956925744221, 10.66776501015650973164761210352

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