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Properties

Label 2-2535-195.134-c0-0-9
Degree $2$
Conductor $2535$
Sign $0.869 - 0.494i$
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  + (1.22 + 0.707i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.707 + 0.707i)5-s + (−0.707 − 1.22i)6-s + (0.499 + 0.866i)9-s + (0.366 + 1.36i)10-s + (0.707 − 1.22i)11-s − 0.999i·12-s + (−0.258 − 0.965i)15-s + (0.499 − 0.866i)16-s + 1.41i·18-s + (−0.258 + 0.965i)20-s + (1.73 − 0.999i)22-s + 1.00i·25-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (0.707 + 0.707i)5-s + (−0.707 − 1.22i)6-s + (0.499 + 0.866i)9-s + (0.366 + 1.36i)10-s + (0.707 − 1.22i)11-s − 0.999i·12-s + (−0.258 − 0.965i)15-s + (0.499 − 0.866i)16-s + 1.41i·18-s + (−0.258 + 0.965i)20-s + (1.73 − 0.999i)22-s + 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.070385169\)
\(L(\frac12)\) \(\approx\) \(2.070385169\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 \)
good2 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.137441710198256029261385802263, −8.020221593024865232824873731239, −7.16807477387348830211334956279, −6.52905083789851032926771758400, −6.05104022465493834143086788287, −5.52360252995575461847249399496, −4.65867459277411677594371297033, −3.66064757833078828388038251897, −2.74172370240954552767481300389, −1.29444201642253026137785762304, 1.40734440720225911588405286562, 2.31877635417026067321426171168, 3.70450563684293059445511879470, 4.30215242366220048687422910510, 5.03015175980883421867997492890, 5.52539962861013759877362402338, 6.36993713722519900926708853852, 7.13279717929258286485680441278, 8.499509198689178883819691303318, 9.282547855773293147673233002967

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