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Properties

Label 2-3276-13.10-c1-0-15
Degree $2$
Conductor $3276$
Sign $0.801 - 0.598i$
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  + 0.118i·5-s + (0.866 + 0.5i)7-s + (3.78 − 2.18i)11-s + (3.33 + 1.38i)13-s + (−3.22 + 5.58i)17-s + (−3.42 − 1.97i)19-s + (−1.34 − 2.32i)23-s + 4.98·25-s + (3.54 + 6.14i)29-s + 8.53i·31-s + (−0.0590 + 0.102i)35-s + (−2.29 + 1.32i)37-s + (1.07 − 0.621i)41-s + (5.45 − 9.45i)43-s + 1.40i·47-s + ⋯
L(s)  = 1  + 0.0528i·5-s + (0.327 + 0.188i)7-s + (1.14 − 0.659i)11-s + (0.923 + 0.382i)13-s + (−0.782 + 1.35i)17-s + (−0.785 − 0.453i)19-s + (−0.280 − 0.485i)23-s + 0.997·25-s + (0.658 + 1.14i)29-s + 1.53i·31-s + (−0.00998 + 0.0172i)35-s + (−0.377 + 0.217i)37-s + (0.168 − 0.0970i)41-s + (0.832 − 1.44i)43-s + 0.204i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.801 - 0.598i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.086682107\)
\(L(\frac12)\) \(\approx\) \(2.086682107\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.33 - 1.38i)T \)
good5 \( 1 - 0.118iT - 5T^{2} \)
11 \( 1 + (-3.78 + 2.18i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.42 + 1.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.34 + 2.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.53iT - 31T^{2} \)
37 \( 1 + (2.29 - 1.32i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.07 + 0.621i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.45 + 9.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.40iT - 47T^{2} \)
53 \( 1 + 5.89T + 53T^{2} \)
59 \( 1 + (-6.96 - 4.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.00 + 5.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.4 + 6.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.58iT - 73T^{2} \)
79 \( 1 + 8.42T + 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + (1.28 - 0.741i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.00126 + 0.000728i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.631778935667068185517628319400, −8.401196312300489699508088601118, −6.90418901283911191465623821025, −6.59557231220891705982122657900, −5.83902011362928566366055181946, −4.78632928878055958000013145574, −4.02019613427899060218104481548, −3.27540386304074469377522445154, −2.01169256507244949046572542182, −1.10175901830411171702111043059, 0.76056358226341084154694272677, 1.88189190615340209571424287909, 2.90808199611483756642358284528, 4.15988554015732749868985836857, 4.42230807827983426709844116211, 5.61206741748597655696230063536, 6.38059545441856087379212682684, 7.01755845663116900435789872265, 7.86294981939487085307116986987, 8.571813807419987176278513784931

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