Properties

Label 2-44e2-44.43-c1-0-13
Degree $2$
Conductor $1936$
Sign $-0.821 - 0.570i$
Analytic cond. $15.4590$
Root an. cond. $3.93179$
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  + 1.73i·3-s + 5-s − 2.44·7-s − 5.65i·13-s + 1.73i·15-s + 4.24i·17-s − 4.24i·21-s + 8.66i·23-s − 4·25-s + 5.19i·27-s + 5.65i·29-s − 1.73i·31-s − 2.44·35-s + 5·37-s + 9.79·39-s + ⋯
L(s)  = 1  + 0.999i·3-s + 0.447·5-s − 0.925·7-s − 1.56i·13-s + 0.447i·15-s + 1.02i·17-s − 0.925i·21-s + 1.80i·23-s − 0.800·25-s + 1.00i·27-s + 1.05i·29-s − 0.311i·31-s − 0.414·35-s + 0.821·37-s + 1.56·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1936\)    =    \(2^{4} \cdot 11^{2}\)
Sign: $-0.821 - 0.570i$
Analytic conductor: \(15.4590\)
Root analytic conductor: \(3.93179\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1936} (1935, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1936,\ (\ :1/2),\ -0.821 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.174539299\)
\(L(\frac12)\) \(\approx\) \(1.174539299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 - 1.73iT - 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8.66iT - 23T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 - 5.65iT - 61T^{2} \)
67 \( 1 - 5.19iT - 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 7.34T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + 17T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.784783976769632728784400610501, −8.950826271567072703486394225620, −7.997073213016561572841351722012, −7.19944534676252116390803966165, −6.02646817155231978692560162836, −5.61717676716492713493174662431, −4.60556714529677619921243665335, −3.54777590705541205081973212234, −3.08598317769891658658228517315, −1.48963669398320285573611783192, 0.41620199703343641056279887952, 1.86922909664973591891713306399, 2.55939272453890653065725893124, 3.88113575883600547995745704780, 4.79879379788926519293429243414, 6.00452260883666275322379211693, 6.75235878594781411827518713107, 6.92115656074618073814936712332, 8.063028473535162325115955604248, 8.863255639004802286583158550152

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