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Properties

Label 2-525-35.27-c1-0-8
Degree $2$
Conductor $525$
Sign $0.915 + 0.403i$
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  + (−0.653 − 0.653i)2-s + (0.707 + 0.707i)3-s − 1.14i·4-s − 0.924i·6-s + (2.53 + 0.741i)7-s + (−2.05 + 2.05i)8-s + 1.00i·9-s − 0.746·11-s + (0.809 − 0.809i)12-s + (4.26 + 4.26i)13-s + (−1.17 − 2.14i)14-s + 0.398·16-s + (5.29 − 5.29i)17-s + (0.653 − 0.653i)18-s + 1.17·19-s + ⋯
L(s)  = 1  + (−0.462 − 0.462i)2-s + (0.408 + 0.408i)3-s − 0.572i·4-s − 0.377i·6-s + (0.959 + 0.280i)7-s + (−0.726 + 0.726i)8-s + 0.333i·9-s − 0.225·11-s + (0.233 − 0.233i)12-s + (1.18 + 1.18i)13-s + (−0.314 − 0.573i)14-s + 0.0996·16-s + (1.28 − 1.28i)17-s + (0.154 − 0.154i)18-s + 0.269·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.915 + 0.403i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42222 - 0.299486i\)
\(L(\frac12)\) \(\approx\) \(1.42222 - 0.299486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.53 - 0.741i)T \)
good2 \( 1 + (0.653 + 0.653i)T + 2iT^{2} \)
11 \( 1 + 0.746T + 11T^{2} \)
13 \( 1 + (-4.26 - 4.26i)T + 13iT^{2} \)
17 \( 1 + (-5.29 + 5.29i)T - 17iT^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + (3.19 - 3.19i)T - 23iT^{2} \)
29 \( 1 + 2.45iT - 29T^{2} \)
31 \( 1 + 4.87iT - 31T^{2} \)
37 \( 1 + (-2.05 - 2.05i)T + 37iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + (-6.64 + 6.64i)T - 43iT^{2} \)
47 \( 1 + (5.29 - 5.29i)T - 47iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 53iT^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 2.52iT - 61T^{2} \)
67 \( 1 + (6.47 + 6.47i)T + 67iT^{2} \)
71 \( 1 + 6.74T + 71T^{2} \)
73 \( 1 + (-2.47 - 2.47i)T + 73iT^{2} \)
79 \( 1 - 5.83iT - 79T^{2} \)
83 \( 1 + (0.768 + 0.768i)T + 83iT^{2} \)
89 \( 1 + 3.69T + 89T^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 97iT^{2} \)
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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

−10.81714389671164287283377078721, −9.778068759571890848737280759966, −9.235732574388925553371260073703, −8.374014995936889988315033810714, −7.46291822904888788918102314119, −5.96572377269128526909937275988, −5.18523935637008326267833532189, −3.99419712401064381507924983566, −2.52430228476155348311307972524, −1.34618464329240115742191662707, 1.26803097964200392410159187899, 3.05533725205019116144450769652, 3.98868515697469473108139497763, 5.55156270587039395362861499060, 6.52331737872818526950334771779, 7.74360264841172031490225023297, 8.102125240497345627351472345741, 8.671539542454668642946737276186, 9.989202547453480200489054740430, 10.80523393178687119531604449096

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