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Properties

Label 2-525-105.53-c1-0-37
Degree $2$
Conductor $525$
Sign $-0.670 + 0.741i$
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  + (0.578 − 2.15i)2-s + (1.56 + 0.737i)3-s + (−2.59 − 1.5i)4-s + (2.5 − 2.95i)6-s + (−0.684 − 2.55i)7-s + (−1.58 + 1.58i)8-s + (1.91 + 2.31i)9-s + (−2.96 − 4.26i)12-s + (−3.74 − 3.74i)13-s − 5.91·14-s + (−0.500 − 0.866i)16-s + (4.31 − 1.15i)17-s + (6.09 − 2.79i)18-s + (1.73 − i)19-s + (0.811 − 4.51i)21-s + ⋯
L(s)  = 1  + (0.409 − 1.52i)2-s + (0.904 + 0.425i)3-s + (−1.29 − 0.750i)4-s + (1.02 − 1.20i)6-s + (−0.258 − 0.965i)7-s + (−0.559 + 0.559i)8-s + (0.637 + 0.770i)9-s + (−0.856 − 1.23i)12-s + (−1.03 − 1.03i)13-s − 1.58·14-s + (−0.125 − 0.216i)16-s + (1.04 − 0.280i)17-s + (1.43 − 0.658i)18-s + (0.397 − 0.229i)19-s + (0.177 − 0.984i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.670 + 0.741i$
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: \((2,\ 525,\ (\ :1/2),\ -0.670 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894325 - 2.01420i\)
\(L(\frac12)\) \(\approx\) \(0.894325 - 2.01420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 - 0.737i)T \)
5 \( 1 \)
7 \( 1 + (0.684 + 2.55i)T \)
good2 \( 1 + (-0.578 + 2.15i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.74 + 3.74i)T + 13iT^{2} \)
17 \( 1 + (-4.31 + 1.15i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.47 - 1.73i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.2 - 2.73i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (1.87 + 1.87i)T + 43iT^{2} \)
47 \( 1 + (2.31 - 8.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.15 + 4.31i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.91 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.684 - 2.55i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-5.11 + 1.36i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.73 + i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.58 - 1.58i)T - 83iT^{2} \)
89 \( 1 + (-2.95 - 5.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.48 - 7.48i)T - 97iT^{2} \)
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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

−10.54924709861756370128040274521, −9.742506700960741538899930776745, −9.404317761861552623422421542607, −7.85806317477038940204470540683, −7.24363572822499486045152820239, −5.26375273837005429513943977421, −4.42967651583522880914147713604, −3.31196593510279501479496793349, −2.80563698804356765437694665974, −1.15755483147547056983736325371, 2.17111134602611000319221239002, 3.56216895646501509585567267820, 4.83850493247961145681195465730, 5.80427578529506890614187294799, 6.77383385414665217479868240265, 7.45916608459692923977394671342, 8.251821968278783912369495514854, 9.168730575272928548869686277327, 9.683590808768060452132953369610, 11.41731671657034725183554165991

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