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Properties

Label 2-525-15.14-c2-0-26
Degree $2$
Conductor $525$
Sign $0.982 + 0.188i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·2-s + (−2.38 − 1.82i)3-s − 2.29·4-s + (−3.11 − 2.38i)6-s + 2.64i·7-s − 8.22·8-s + (2.35 + 8.68i)9-s + 2.61i·11-s + (5.45 + 4.17i)12-s − 6.35i·13-s + 3.45i·14-s − 1.58·16-s + 12.1·17-s + (3.07 + 11.3i)18-s + 10.2·19-s + ⋯
L(s)  = 1  + 0.653·2-s + (−0.794 − 0.607i)3-s − 0.572·4-s + (−0.519 − 0.397i)6-s + 0.377i·7-s − 1.02·8-s + (0.261 + 0.965i)9-s + 0.237i·11-s + (0.454 + 0.348i)12-s − 0.488i·13-s + 0.247i·14-s − 0.0989·16-s + 0.714·17-s + (0.170 + 0.630i)18-s + 0.538·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.982 + 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.421116597\)
\(L(\frac12)\) \(\approx\) \(1.421116597\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.38 + 1.82i)T \)
5 \( 1 \)
7 \( 1 - 2.64iT \)
good2 \( 1 - 1.30T + 4T^{2} \)
11 \( 1 - 2.61iT - 121T^{2} \)
13 \( 1 + 6.35iT - 169T^{2} \)
17 \( 1 - 12.1T + 289T^{2} \)
19 \( 1 - 10.2T + 361T^{2} \)
23 \( 1 - 4.30T + 529T^{2} \)
29 \( 1 - 17.3iT - 841T^{2} \)
31 \( 1 - 39.2T + 961T^{2} \)
37 \( 1 + 41.0iT - 1.36e3T^{2} \)
41 \( 1 - 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8iT - 1.84e3T^{2} \)
47 \( 1 - 39.9T + 2.20e3T^{2} \)
53 \( 1 - 105.T + 2.80e3T^{2} \)
59 \( 1 - 41.3iT - 3.48e3T^{2} \)
61 \( 1 + 20.4T + 3.72e3T^{2} \)
67 \( 1 - 27.1iT - 4.48e3T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 - 60.7iT - 5.32e3T^{2} \)
79 \( 1 - 63.2T + 6.24e3T^{2} \)
83 \( 1 - 89.9T + 6.88e3T^{2} \)
89 \( 1 - 63.1iT - 7.92e3T^{2} \)
97 \( 1 + 19.1iT - 9.40e3T^{2} \)
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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

−10.72172952655185841771163478899, −9.864318822124577785180987415288, −8.787216803100350080542385617952, −7.82154378939451184580430980659, −6.79524275963082522492740541981, −5.65187151806620372905992496162, −5.25625148357808807550712251457, −4.04888818447107646852848341439, −2.66880784293618619556696708474, −0.866848663763814196456229297777, 0.78599318206106636357645982164, 3.17327799657007449077573910942, 4.14567559639081434435758691615, 4.92614378384085033526829787551, 5.81764481927230000392305104236, 6.67472077336989478622845966624, 8.017560198640517302268656059874, 9.163272001706818688673851176145, 9.841610928299728130553858968653, 10.67617688352746242165149670337

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