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Properties

Label 2-525-5.4-c3-0-11
Degree $2$
Conductor $525$
Sign $-0.894 - 0.447i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.217i·2-s + 3i·3-s + 7.95·4-s − 0.652·6-s + 7i·7-s + 3.46i·8-s − 9·9-s − 30.6·11-s + 23.8i·12-s − 25.3i·13-s − 1.52·14-s + 62.8·16-s + 72.8i·17-s − 1.95i·18-s − 122.·19-s + ⋯
L(s)  = 1  + 0.0768i·2-s + 0.577i·3-s + 0.994·4-s − 0.0443·6-s + 0.377i·7-s + 0.153i·8-s − 0.333·9-s − 0.838·11-s + 0.573i·12-s − 0.540i·13-s − 0.0290·14-s + 0.982·16-s + 1.03i·17-s − 0.0256i·18-s − 1.48·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.383421145\)
\(L(\frac12)\) \(\approx\) \(1.383421145\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good2 \( 1 - 0.217iT - 8T^{2} \)
11 \( 1 + 30.6T + 1.33e3T^{2} \)
13 \( 1 + 25.3iT - 2.19e3T^{2} \)
17 \( 1 - 72.8iT - 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 - 194. iT - 1.21e4T^{2} \)
29 \( 1 + 48.6T + 2.43e4T^{2} \)
31 \( 1 + 288.T + 2.97e4T^{2} \)
37 \( 1 - 15.8iT - 5.06e4T^{2} \)
41 \( 1 - 452.T + 6.89e4T^{2} \)
43 \( 1 - 152. iT - 7.95e4T^{2} \)
47 \( 1 + 164. iT - 1.03e5T^{2} \)
53 \( 1 - 591. iT - 1.48e5T^{2} \)
59 \( 1 - 180.T + 2.05e5T^{2} \)
61 \( 1 - 115.T + 2.26e5T^{2} \)
67 \( 1 + 605. iT - 3.00e5T^{2} \)
71 \( 1 + 990.T + 3.57e5T^{2} \)
73 \( 1 - 863. iT - 3.89e5T^{2} \)
79 \( 1 + 965.T + 4.93e5T^{2} \)
83 \( 1 - 160. iT - 5.71e5T^{2} \)
89 \( 1 - 51.6T + 7.04e5T^{2} \)
97 \( 1 + 1.49e3iT - 9.12e5T^{2} \)
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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

−10.82758697001819348110386030324, −10.16912878408953148553777739115, −9.065090532102588958724018681799, −8.065845301719561996733667690675, −7.31784103169678455689910521733, −5.97782954656834583369993148962, −5.52212302474173555181323913114, −4.02708928310756502997569174258, −2.90745631416663258478469941981, −1.79744278634277509679611888637, 0.36786934100420709042646277334, 1.96365098473575479591999755737, 2.76136498128472708668165739882, 4.24642887871162799198089156343, 5.58608329344253983072368049537, 6.60611690079477015569611944297, 7.22108435965952778108644331224, 8.085256543639186192333306757339, 9.118859908188977606843158504214, 10.41491687873557548834397987226

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