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Properties

Label 2-3276-13.10-c1-0-18
Degree $2$
Conductor $3276$
Sign $0.979 + 0.202i$
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  + 1.40i·5-s + (−0.866 − 0.5i)7-s + (1.03 − 0.596i)11-s + (−1.17 − 3.40i)13-s + (−2.57 + 4.45i)17-s + (−1.13 − 0.656i)19-s + (−0.660 − 1.14i)23-s + 3.02·25-s + (2.70 + 4.68i)29-s − 7.30i·31-s + (0.702 − 1.21i)35-s + (7.44 − 4.29i)37-s + (2.44 − 1.41i)41-s + (−0.343 + 0.595i)43-s + 1.80i·47-s + ⋯
L(s)  = 1  + 0.628i·5-s + (−0.327 − 0.188i)7-s + (0.311 − 0.179i)11-s + (−0.326 − 0.945i)13-s + (−0.624 + 1.08i)17-s + (−0.260 − 0.150i)19-s + (−0.137 − 0.238i)23-s + 0.604·25-s + (0.501 + 0.869i)29-s − 1.31i·31-s + (0.118 − 0.205i)35-s + (1.22 − 0.706i)37-s + (0.381 − 0.220i)41-s + (−0.0524 + 0.0908i)43-s + 0.263i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.979 + 0.202i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.632440168\)
\(L(\frac12)\) \(\approx\) \(1.632440168\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 3.40i)T \)
good5 \( 1 - 1.40iT - 5T^{2} \)
11 \( 1 + (-1.03 + 0.596i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.57 - 4.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 + 0.656i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.660 + 1.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.70 - 4.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.30iT - 31T^{2} \)
37 \( 1 + (-7.44 + 4.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.44 + 1.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.343 - 0.595i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.80iT - 47T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + (-2.25 - 1.30i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0222 + 0.0384i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.14 - 1.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.94 - 1.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 7.59T + 79T^{2} \)
83 \( 1 + 8.92iT - 83T^{2} \)
89 \( 1 + (5.99 - 3.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.4 - 6.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.606180037309770061053118463192, −7.82954448761381091095592795937, −7.11772076767434744026075723163, −6.32295078574522131764073273114, −5.81438308335104434561401050242, −4.68082873666588341491437816972, −3.86857858350648198935449323644, −3.00446825411053676108330810167, −2.16299769136513202674803282273, −0.67558772767604407609006503910, 0.856568848229762355328456114754, 2.08815127034196016917504354922, 2.99262030196989116275496682641, 4.20947854682877491449184565037, 4.71346424313911051548876595710, 5.57189613350865831965933958778, 6.61142310098924026986528921540, 6.97575161295096860848413208151, 8.041631419669559932027910148689, 8.760016907597191260126623343442

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