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Properties

Label 2-2535-15.14-c0-0-15
Degree $2$
Conductor $2535$
Sign $-0.707 + 0.707i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − i·3-s + 1.00·4-s + (−0.707 − 0.707i)5-s − 1.41i·6-s − 9-s + (−1.00 − 1.00i)10-s − 1.41i·11-s − 1.00i·12-s + (−0.707 + 0.707i)15-s − 0.999·16-s − 1.41·18-s + (−0.707 − 0.707i)20-s − 2.00i·22-s + 1.00i·25-s + ⋯
L(s)  = 1  + 1.41·2-s − i·3-s + 1.00·4-s + (−0.707 − 0.707i)5-s − 1.41i·6-s − 9-s + (−1.00 − 1.00i)10-s − 1.41i·11-s − 1.00i·12-s + (−0.707 + 0.707i)15-s − 0.999·16-s − 1.41·18-s + (−0.707 − 0.707i)20-s − 2.00i·22-s + 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.790827183\)
\(L(\frac12)\) \(\approx\) \(1.790827183\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 \)
good2 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.695955262070830955908971478406, −7.913943695328618217858419281277, −7.13391649363453096638784455444, −6.25108017378555535065554633810, −5.62433852537949346081667969346, −4.98498336554112304878509088707, −3.87421608395209331126094681757, −3.30412349908758382422667165251, −2.23707341718821735702670494180, −0.71993469791529272885166086389, 2.38785000461854475326598300241, 3.08891906955447125221840340717, 4.08346817964499020483801640221, 4.36545196550477090025509194084, 5.22907091750044998805372023082, 6.05914896566363860179604439391, 6.90034848369628328557204696554, 7.62026838775877530834954974897, 8.669878901105389132230483185621, 9.502888876469921803097051082753

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