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Properties

Label 2-3276-13.10-c1-0-11
Degree $2$
Conductor $3276$
Sign $0.861 - 0.507i$
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  − 2.36i·5-s + (−0.866 − 0.5i)7-s + (−2.56 + 1.47i)11-s + (3.15 + 1.73i)13-s + (−1.25 + 2.17i)17-s + (3.52 + 2.03i)19-s + (1.45 + 2.52i)23-s − 0.583·25-s + (0.464 + 0.805i)29-s − 2.20i·31-s + (−1.18 + 2.04i)35-s + (−5.73 + 3.31i)37-s + (−9.87 + 5.69i)41-s + (0.859 − 1.48i)43-s − 2.64i·47-s + ⋯
L(s)  = 1  − 1.05i·5-s + (−0.327 − 0.188i)7-s + (−0.772 + 0.445i)11-s + (0.876 + 0.481i)13-s + (−0.303 + 0.526i)17-s + (0.808 + 0.466i)19-s + (0.304 + 0.526i)23-s − 0.116·25-s + (0.0863 + 0.149i)29-s − 0.396i·31-s + (−0.199 + 0.345i)35-s + (−0.942 + 0.544i)37-s + (−1.54 + 0.890i)41-s + (0.131 − 0.226i)43-s − 0.385i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.861 - 0.507i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517602942\)
\(L(\frac12)\) \(\approx\) \(1.517602942\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 - 1.73i)T \)
good5 \( 1 + 2.36iT - 5T^{2} \)
11 \( 1 + (2.56 - 1.47i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.25 - 2.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.52 - 2.03i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 2.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.464 - 0.805i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.20iT - 31T^{2} \)
37 \( 1 + (5.73 - 3.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.87 - 5.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.859 + 1.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.64iT - 47T^{2} \)
53 \( 1 - 0.0492T + 53T^{2} \)
59 \( 1 + (-7.11 - 4.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.48 - 2.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.51 + 3.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.58 - 4.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.56iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (5.06 - 2.92i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.5 - 8.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.550589451823677279469312517676, −8.207712035068102579951249892122, −7.20139856642729197534069674366, −6.49808088970937589491604123306, −5.49336637745443433660015670123, −4.98412104189843539584771123524, −4.04032721904989294427985219578, −3.26677237929679349368457075426, −1.94158539861616779490488882816, −1.00841404274724712868692172284, 0.55078653879224086544511444314, 2.16908553815286596903041309933, 3.11477987560096763830107778213, 3.49359905914387186004153597705, 4.89570015167530618405313347494, 5.55939198821869460888492738943, 6.47368791414788409439881754575, 6.97812710984447240361241860628, 7.78992021168815372110939871327, 8.591469207264023184291980220844

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