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Properties

Label 2-525-35.3-c1-0-11
Degree $2$
Conductor $525$
Sign $0.165 + 0.986i$
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  + (−2.17 − 0.582i)2-s + (0.258 + 0.965i)3-s + (2.65 + 1.53i)4-s − 2.25i·6-s + (0.660 − 2.56i)7-s + (−1.69 − 1.69i)8-s + (−0.866 + 0.499i)9-s + (0.329 − 0.571i)11-s + (−0.793 + 2.96i)12-s + (2.55 − 2.55i)13-s + (−2.92 + 5.18i)14-s + (−0.366 − 0.635i)16-s + (−5.43 + 1.45i)17-s + (2.17 − 0.582i)18-s + (−1.48 − 2.56i)19-s + ⋯
L(s)  = 1  + (−1.53 − 0.411i)2-s + (0.149 + 0.557i)3-s + (1.32 + 0.766i)4-s − 0.918i·6-s + (0.249 − 0.968i)7-s + (−0.599 − 0.599i)8-s + (−0.288 + 0.166i)9-s + (0.0994 − 0.172i)11-s + (−0.229 + 0.854i)12-s + (0.709 − 0.709i)13-s + (−0.782 + 1.38i)14-s + (−0.0916 − 0.158i)16-s + (−1.31 + 0.353i)17-s + (0.512 − 0.137i)18-s + (−0.339 − 0.588i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467565 - 0.395615i\)
\(L(\frac12)\) \(\approx\) \(0.467565 - 0.395615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-0.660 + 2.56i)T \)
good2 \( 1 + (2.17 + 0.582i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.329 + 0.571i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.55 + 2.55i)T - 13iT^{2} \)
17 \( 1 + (5.43 - 1.45i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.48 + 2.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0726 - 0.271i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.03iT - 29T^{2} \)
31 \( 1 + (-6.53 - 3.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.79 - 2.08i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (8.53 + 8.53i)T + 43iT^{2} \)
47 \( 1 + (-3.11 + 11.6i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.65 + 1.24i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.782 - 1.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.02 - 0.589i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.762 + 2.84i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.13T + 71T^{2} \)
73 \( 1 + (0.417 + 1.55i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.17 + 3.56i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.14 + 2.14i)T - 83iT^{2} \)
89 \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.33 - 3.33i)T + 97iT^{2} \)
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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

−10.47794830193070653585806144473, −9.954560163588106905411849995170, −8.763053477654047692850031949603, −8.424639964688152515703684095406, −7.37694138147241499065243277872, −6.40305784040252332265666985503, −4.80051198740170674315796828188, −3.65188169716050868332233771290, −2.22498410529389150241209065356, −0.62309203590198749152799597534, 1.41921176704633257426617030541, 2.53051884405103189214153323014, 4.48382741227447665372967323107, 6.14858754063295624166815217770, 6.58794513995598784051063582095, 7.76288539364970537344340732039, 8.441431827716810565804533789596, 9.095534581662576414790361161860, 9.785936355024122349443912587374, 11.08578642882693312134017451759

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