Properties

Label 2-60e2-15.14-c2-0-45
Degree $2$
Conductor $3600$
Sign $0.472 + 0.881i$
Analytic cond. $98.0928$
Root an. cond. $9.90418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·7-s + 16.9i·11-s − 8i·13-s − 12.7·17-s − 16·19-s − 16.9·23-s + 4.24i·29-s − 44·31-s − 34i·37-s − 46.6i·41-s − 40i·43-s + 84.8·47-s + 33·49-s − 38.1·53-s − 33.9i·59-s + ⋯
L(s)  = 1  + 0.571i·7-s + 1.54i·11-s − 0.615i·13-s − 0.748·17-s − 0.842·19-s − 0.737·23-s + 0.146i·29-s − 1.41·31-s − 0.918i·37-s − 1.13i·41-s − 0.930i·43-s + 1.80·47-s + 0.673·49-s − 0.720·53-s − 0.575i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.472 + 0.881i$
Analytic conductor: \(98.0928\)
Root analytic conductor: \(9.90418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1),\ 0.472 + 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.107117711\)
\(L(\frac12)\) \(\approx\) \(1.107117711\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4iT - 49T^{2} \)
11 \( 1 - 16.9iT - 121T^{2} \)
13 \( 1 + 8iT - 169T^{2} \)
17 \( 1 + 12.7T + 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 + 16.9T + 529T^{2} \)
29 \( 1 - 4.24iT - 841T^{2} \)
31 \( 1 + 44T + 961T^{2} \)
37 \( 1 + 34iT - 1.36e3T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 40iT - 1.84e3T^{2} \)
47 \( 1 - 84.8T + 2.20e3T^{2} \)
53 \( 1 + 38.1T + 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 - 50T + 3.72e3T^{2} \)
67 \( 1 + 8iT - 4.48e3T^{2} \)
71 \( 1 + 50.9iT - 5.04e3T^{2} \)
73 \( 1 - 16iT - 5.32e3T^{2} \)
79 \( 1 + 76T + 6.24e3T^{2} \)
83 \( 1 - 118.T + 6.88e3T^{2} \)
89 \( 1 - 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 176iT - 9.40e3T^{2} \)
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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

−8.294260703176643198806978171045, −7.39263627608952614784292180794, −6.93454743640997610207329824872, −5.90895261288427929672620154760, −5.30072665077767955990513057722, −4.37292885685795860683821094097, −3.70208875443575326618600155733, −2.30777988853573116714202281444, −1.98079620785250773521164167292, −0.28764384775761840503409300458, 0.815549620312793193444007062190, 1.97235100374910980055079660372, 3.02495812600685257221240920940, 3.94155193419132361395351100578, 4.51748468176779986932623218938, 5.67073664775078830912228929074, 6.25192813680709348381649008062, 6.97601354579089690894313090741, 7.82720586428621170746411707581, 8.565095037515012308826471427369

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