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Properties

Label 2-525-35.33-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.625 + 0.780i$
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  + (0.698 + 2.60i)2-s + (0.965 + 0.258i)3-s + (−4.57 + 2.64i)4-s + 2.69i·6-s + (−2.58 + 0.543i)7-s + (−6.26 − 6.26i)8-s + (0.866 + 0.499i)9-s + (1.69 + 2.93i)11-s + (−5.10 + 1.36i)12-s + (−4.01 + 4.01i)13-s + (−3.22 − 6.36i)14-s + (6.67 − 11.5i)16-s + (1.00 − 3.76i)17-s + (−0.698 + 2.60i)18-s + (0.721 − 1.24i)19-s + ⋯
L(s)  = 1  + (0.493 + 1.84i)2-s + (0.557 + 0.149i)3-s + (−2.28 + 1.32i)4-s + 1.10i·6-s + (−0.978 + 0.205i)7-s + (−2.21 − 2.21i)8-s + (0.288 + 0.166i)9-s + (0.511 + 0.885i)11-s + (−1.47 + 0.394i)12-s + (−1.11 + 1.11i)13-s + (−0.862 − 1.70i)14-s + (1.66 − 2.89i)16-s + (0.244 − 0.911i)17-s + (−0.164 + 0.614i)18-s + (0.165 − 0.286i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.625 + 0.780i$
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: \((2,\ 525,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592574 - 1.23356i\)
\(L(\frac12)\) \(\approx\) \(0.592574 - 1.23356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 - 0.543i)T \)
good2 \( 1 + (-0.698 - 2.60i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.69 - 2.93i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.01 - 4.01i)T - 13iT^{2} \)
17 \( 1 + (-1.00 + 3.76i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.721 + 1.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.739 - 0.198i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-3.17 + 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.325 - 1.21i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.05iT - 41T^{2} \)
43 \( 1 + (-6.74 - 6.74i)T + 43iT^{2} \)
47 \( 1 + (6.55 - 1.75i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.94 - 10.9i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.22 - 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.9 - 6.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.544 - 0.145i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 + (-4.78 - 1.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.79 + 3.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.49 + 1.49i)T - 83iT^{2} \)
89 \( 1 + (-1.78 + 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.36 + 1.36i)T + 97iT^{2} \)
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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

−11.87487395995274362196220046192, −9.809857779383420451778870513518, −9.432476322956024413547484304507, −8.688361998422388887916383585349, −7.33362969797578974724777640419, −7.10097700476864518618406457128, −6.10585330488872159540604969799, −4.88157032086746420133436069295, −4.20132685013082575204813532791, −2.89006779820686701070590493752, 0.64733789715536205741436425982, 2.28346273830527611101506384108, 3.28810226773975133401336584557, 3.89338296434356930605646360415, 5.26669717190730851797607354188, 6.31339809599703530093583404542, 7.957339404106697016748777480736, 8.873881107799847143480766192580, 9.948113078197249282606074172777, 10.10451170262489836960886117227

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