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Properties

Label 2-525-35.12-c1-0-4
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: $\chi_{525} (82, \cdot )$
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

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

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