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Properties

Label 2-3648-76.75-c1-0-61
Degree $2$
Conductor $3648$
Sign $0.917 + 0.397i$
Analytic cond. $29.1294$
Root an. cond. $5.39716$
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-s + 3.37·5-s + 0.792i·7-s + 9-s − 0.792i·11-s − 5.04i·13-s + 3.37·15-s + 5.37·17-s + (4 + 1.73i)19-s + 0.792i·21-s − 8.51i·23-s + 6.37·25-s + 27-s + 10.0i·29-s − 8.74·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.50·5-s + 0.299i·7-s + 0.333·9-s − 0.238i·11-s − 1.40i·13-s + 0.870·15-s + 1.30·17-s + (0.917 + 0.397i)19-s + 0.172i·21-s − 1.77i·23-s + 1.27·25-s + 0.192·27-s + 1.87i·29-s − 1.57·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.917 + 0.397i$
Analytic conductor: \(29.1294\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :1/2),\ 0.917 + 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.421376473\)
\(L(\frac12)\) \(\approx\) \(3.421376473\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 - 0.792iT - 7T^{2} \)
11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 + 5.04iT - 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
37 \( 1 + 5.04iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 + 4.25iT - 47T^{2} \)
53 \( 1 + 3.16iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 + 9.48T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 4.11T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−8.699896021305508523859984832778, −7.75195263991732984061620153465, −7.12145597092322887159557171932, −6.00895788427799988684824332662, −5.58463534580429075960284593967, −4.95770600804308846737575070801, −3.49407025271866895549336362909, −2.93618501946777022304765433476, −2.00195793452515717311560927867, −1.01577913008294592744260948104, 1.35997600821828411169129336213, 1.93327150470503665411265597743, 2.97668647223354250702028465294, 3.85812714181121875707816379990, 4.85179618731856661595905453243, 5.69227700114511554650873660263, 6.23505791514068290164161234119, 7.37102068236657129349117195587, 7.56183338110888986234400348889, 8.881825797617804810171233510423

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