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Properties

Label 2-3276-13.10-c1-0-0
Degree $2$
Conductor $3276$
Sign $-0.998 - 0.0599i$
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  + 1.38i·5-s + (0.866 + 0.5i)7-s + (−0.998 + 0.576i)11-s + (−1.65 − 3.20i)13-s + (0.781 − 1.35i)17-s + (−7.26 − 4.19i)19-s + (1.11 + 1.92i)23-s + 3.08·25-s + (1.43 + 2.48i)29-s + 6.65i·31-s + (−0.692 + 1.19i)35-s + (3.01 − 1.74i)37-s + (−10.8 + 6.27i)41-s + (−3.07 + 5.33i)43-s + 7.40i·47-s + ⋯
L(s)  = 1  + 0.619i·5-s + (0.327 + 0.188i)7-s + (−0.301 + 0.173i)11-s + (−0.458 − 0.888i)13-s + (0.189 − 0.328i)17-s + (−1.66 − 0.962i)19-s + (0.231 + 0.401i)23-s + 0.616·25-s + (0.266 + 0.461i)29-s + 1.19i·31-s + (−0.117 + 0.202i)35-s + (0.495 − 0.286i)37-s + (−1.69 + 0.980i)41-s + (−0.469 + 0.813i)43-s + 1.08i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.998 - 0.0599i$
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: \((2,\ 3276,\ (\ :1/2),\ -0.998 - 0.0599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3135094543\)
\(L(\frac12)\) \(\approx\) \(0.3135094543\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 3.20i)T \)
good5 \( 1 - 1.38iT - 5T^{2} \)
11 \( 1 + (0.998 - 0.576i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.781 + 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.26 + 4.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 - 1.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.43 - 2.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.65iT - 31T^{2} \)
37 \( 1 + (-3.01 + 1.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.8 - 6.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.07 - 5.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.40iT - 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (6.97 + 4.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.829 - 1.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.21 - 1.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.60 + 4.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.30iT - 73T^{2} \)
79 \( 1 - 4.64T + 79T^{2} \)
83 \( 1 + 1.82iT - 83T^{2} \)
89 \( 1 + (2.05 - 1.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.53 + 3.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.926767644387829563726759054751, −8.239978770492589758561182778491, −7.52101324353806833253211088844, −6.75275297428382288283162335305, −6.14607067665516878720371293935, −4.96911154923371773676514885196, −4.67200401978827975416721952345, −3.18146139637107332950062514206, −2.76328988808879524712243239636, −1.53199388859102603460161234296, 0.090358064393928066624774632116, 1.56150463914353849337620011411, 2.37735660485532666024508841213, 3.71183574604721569907754868601, 4.43635139730649648996983957604, 5.08560788182521583842170104695, 6.06912843886249595157109268313, 6.71840576511552826100738118483, 7.64688395988022864876898871930, 8.422787289343644551864277725347

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