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Properties

Label 2-525-35.3-c1-0-12
Degree $2$
Conductor $525$
Sign $0.296 - 0.955i$
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  + (2.01 + 0.538i)2-s + (0.258 + 0.965i)3-s + (2.02 + 1.16i)4-s + 2.08i·6-s + (1.99 + 1.74i)7-s + (0.495 + 0.495i)8-s + (−0.866 + 0.499i)9-s + (−0.971 + 1.68i)11-s + (−0.604 + 2.25i)12-s + (−1.84 + 1.84i)13-s + (3.06 + 4.57i)14-s + (−1.60 − 2.78i)16-s + (6.06 − 1.62i)17-s + (−2.01 + 0.538i)18-s + (1.50 + 2.61i)19-s + ⋯
L(s)  = 1  + (1.42 + 0.381i)2-s + (0.149 + 0.557i)3-s + (1.01 + 0.584i)4-s + 0.850i·6-s + (0.752 + 0.658i)7-s + (0.175 + 0.175i)8-s + (−0.288 + 0.166i)9-s + (−0.292 + 0.507i)11-s + (−0.174 + 0.651i)12-s + (−0.511 + 0.511i)13-s + (0.819 + 1.22i)14-s + (−0.401 − 0.695i)16-s + (1.47 − 0.394i)17-s + (−0.474 + 0.127i)18-s + (0.345 + 0.599i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.296 - 0.955i$
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: \((2,\ 525,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56304 + 1.88882i\)
\(L(\frac12)\) \(\approx\) \(2.56304 + 1.88882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 - 1.74i)T \)
good2 \( 1 + (-2.01 - 0.538i)T + (1.73 + i)T^{2} \)
11 \( 1 + (0.971 - 1.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 1.84i)T - 13iT^{2} \)
17 \( 1 + (-6.06 + 1.62i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.50 - 2.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.43 + 5.35i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 + (5.10 + 2.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.901 + 0.241i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (6.54 + 6.54i)T + 43iT^{2} \)
47 \( 1 + (-1.00 + 3.75i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.12 + 0.301i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.06 - 5.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.85 + 14.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + (1.62 + 6.05i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (14.2 - 8.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.197 + 0.197i)T - 83iT^{2} \)
89 \( 1 + (6.08 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \)
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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

−11.41992504236023889649649968279, −10.12317797962935704222447985072, −9.383293246393509629298487536578, −8.161518805899756156516450755707, −7.28835485917245072449600135836, −6.05290461738530794140022237921, −5.19682647216397904046913584158, −4.61527564055007483081622654661, −3.48331889347045042680027645857, −2.30806247234526183918234409537, 1.45135560670990480191187602897, 2.95720969934421013083676605284, 3.76050687056584418854750347644, 5.16476272061972096376308202235, 5.55755861511720162354238684422, 6.99089737202005151367394650446, 7.74846930460905029711040261718, 8.753241276617032610103690920076, 10.14965364585292611787372659213, 11.05501821692500168005453483727

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