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Properties

Label 2-2535-15.14-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5488803221\)
\(L(\frac12)\) \(\approx\) \(0.5488803221\)
\(L(1)\) 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

−9.187671091775549649479953253714, −8.413525898723746181857348781475, −7.61360695861390972620116993206, −7.08819540457504688088423284780, −6.56425533009804021082687767776, −5.63104421706273913821364831360, −4.50001964969558968973949130297, −2.91664872133475514087610492399, −2.08215652076035778207754203795, −1.36076000368243166167401092371, 0.60837504971006604754676140524, 1.96272916721019942757783598776, 3.14229826521066610074728408985, 4.23349110043376554986890534056, 5.18796969046936172675526983443, 5.86267222346859585597791650525, 6.75150933822443521353306807154, 8.029430009888972895668980551724, 8.444643695836040841439053189467, 9.101684597539623477507508585653

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