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Properties

Label 2-2527-7.6-c0-0-11
Degree $2$
Conductor $2527$
Sign $i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + 2-s − i·3-s + i·5-s − i·6-s − i·7-s − 8-s + i·10-s − i·13-s − i·14-s + 15-s − 16-s − i·17-s − 21-s + 23-s + i·24-s + ⋯
L(s)  = 1  + 2-s − i·3-s + i·5-s − i·6-s − i·7-s − 8-s + i·10-s − i·13-s − i·14-s + 15-s − 16-s − i·17-s − 21-s + 23-s + i·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.667135843\)
\(L(\frac12)\) \(\approx\) \(1.667135843\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
19 \( 1 \)
good2 \( 1 - T + T^{2} \)
3 \( 1 + iT - T^{2} \)
5 \( 1 - iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 - iT - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−8.821686309008154509861791287902, −7.71631281296287214951557952153, −7.19192108893892525163257030331, −6.71251130313808231054164231790, −5.80352634358858358876501688478, −4.99393567371456382548914178515, −3.99973783900039567587519135055, −3.21107693290664982489490744863, −2.43857261917122828201239278730, −0.836682825845883495738864478565, 1.71815104399877751179121851700, 3.06728027885089978539652377117, 3.92466994099978142739440660449, 4.65850300858125860409942080219, 5.03728962431829931348296105397, 5.85969268866724301581022314486, 6.60206188980144729955164522794, 7.961769119485148725221980259775, 8.885363211521434274832190037052, 9.203670979821990706683090484286

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