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Properties

Label 2-3276-13.10-c1-0-17
Degree $2$
Conductor $3276$
Sign $0.989 + 0.141i$
Analytic cond. $26.1589$
Root an. cond. $5.11458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.14i·5-s + (0.866 + 0.5i)7-s + (3.33 − 1.92i)11-s + (1.38 + 3.32i)13-s + (−0.0971 + 0.168i)17-s + (5.33 + 3.07i)19-s + (4.01 + 6.94i)23-s − 4.90·25-s + (−1.35 − 2.35i)29-s + 8.51i·31-s + (1.57 − 2.72i)35-s + (3.10 − 1.79i)37-s + (3.44 − 1.99i)41-s + (−5.74 + 9.95i)43-s + 4.35i·47-s + ⋯
L(s)  = 1  − 1.40i·5-s + (0.327 + 0.188i)7-s + (1.00 − 0.580i)11-s + (0.384 + 0.923i)13-s + (−0.0235 + 0.0408i)17-s + (1.22 + 0.706i)19-s + (0.836 + 1.44i)23-s − 0.981·25-s + (−0.252 − 0.437i)29-s + 1.52i·31-s + (0.266 − 0.460i)35-s + (0.511 − 0.295i)37-s + (0.538 − 0.311i)41-s + (−0.876 + 1.51i)43-s + 0.634i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(26.1589\)
Root analytic conductor: \(5.11458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3276} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3276,\ (\ :1/2),\ 0.989 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.242870183\)
\(L(\frac12)\) \(\approx\) \(2.242870183\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-1.38 - 3.32i)T \)
good5 \( 1 + 3.14iT - 5T^{2} \)
11 \( 1 + (-3.33 + 1.92i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0971 - 0.168i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.33 - 3.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.01 - 6.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.35 + 2.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.51iT - 31T^{2} \)
37 \( 1 + (-3.10 + 1.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.44 + 1.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.74 - 9.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.35iT - 47T^{2} \)
53 \( 1 - 0.576T + 53T^{2} \)
59 \( 1 + (1.80 + 1.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.70 - 2.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.69 + 2.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.92 - 4.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.31iT - 73T^{2} \)
79 \( 1 + 8.98T + 79T^{2} \)
83 \( 1 + 7.66iT - 83T^{2} \)
89 \( 1 + (4.42 - 2.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.33 - 4.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.714964174075323107581032325181, −8.042163426759311503354950521975, −7.21563134988285551821101723357, −6.24627227884383944609314148943, −5.50469400353323069268144999860, −4.83692842755744534087183144424, −4.00837389509533188062857958993, −3.19916905125993516393158873821, −1.53309882914458696878732677052, −1.16790926057871053960622291598, 0.847337988019913847086982305967, 2.24034374935080939787102805375, 3.06115082831815296371044343173, 3.82768639505690434450177319138, 4.79520244474905358892425627025, 5.73188354676495501118624959444, 6.63694124501109282520162937190, 7.05366417356949787865108020458, 7.75423580668816341457687441870, 8.638932594880311374996899004329

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