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Properties

Label 2-525-105.53-c1-0-3
Degree $2$
Conductor $525$
Sign $-0.997 + 0.0688i$
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  + (−0.340 + 1.26i)2-s + (−0.224 − 1.71i)3-s + (0.236 + 0.136i)4-s + (2.25 + 0.299i)6-s + (−2.32 + 1.25i)7-s + (−2.11 + 2.11i)8-s + (−2.89 + 0.769i)9-s + (−3.38 − 1.95i)11-s + (0.181 − 0.436i)12-s + (1.56 + 1.56i)13-s + (−0.807 − 3.38i)14-s + (−1.69 − 2.92i)16-s + (−2.58 + 0.693i)17-s + (0.00887 − 3.94i)18-s + (−1.61 + 0.930i)19-s + ⋯
L(s)  = 1  + (−0.240 + 0.897i)2-s + (−0.129 − 0.991i)3-s + (0.118 + 0.0681i)4-s + (0.921 + 0.122i)6-s + (−0.879 + 0.476i)7-s + (−0.746 + 0.746i)8-s + (−0.966 + 0.256i)9-s + (−1.01 − 0.588i)11-s + (0.0523 − 0.125i)12-s + (0.434 + 0.434i)13-s + (−0.215 − 0.903i)14-s + (−0.422 − 0.731i)16-s + (−0.627 + 0.168i)17-s + (0.00209 − 0.929i)18-s + (−0.369 + 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.997 + 0.0688i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.997 + 0.0688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0113026 - 0.327953i\)
\(L(\frac12)\) \(\approx\) \(0.0113026 - 0.327953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.224 + 1.71i)T \)
5 \( 1 \)
7 \( 1 + (2.32 - 1.25i)T \)
good2 \( 1 + (0.340 - 1.26i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (2.58 - 0.693i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.61 - 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 0.638i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.513T + 29T^{2} \)
31 \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.60 - 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (7.60 + 7.60i)T + 43iT^{2} \)
47 \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.498 - 1.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.259 - 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.34 + 8.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (2.79 - 0.749i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \)
89 \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.81 + 6.81i)T - 97iT^{2} \)
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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

−11.40293901031081724623804219263, −10.53395887328987732132280034885, −9.065666677693792294496066948418, −8.449992179071592831725296882513, −7.65320811195995684440132837276, −6.65616804053916131243573624700, −6.16824915568954099228068766804, −5.28933278382090763312648276315, −3.25517658577993861269543304393, −2.20067169253495732133525350917, 0.18811273569908813014108717593, 2.38379043650658358146384070178, 3.37730493662064770975594357463, 4.37036790288760557479838029532, 5.70047920966088549471760909317, 6.57972331652556699580350957907, 7.85931646202411564618620218528, 9.150588809289465920734808619437, 9.784872965990101663477759950006, 10.43051192545286883691323135210

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