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Properties

Label 2-60e2-15.14-c0-0-2
Degree $2$
Conductor $3600$
Sign $0.881 + 0.472i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·7-s − 1.41i·11-s − i·13-s + 1.41·17-s + 19-s − 1.41·23-s − 1.41i·29-s − 31-s + i·43-s + 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s + 1.41·83-s + ⋯
L(s)  = 1  + i·7-s − 1.41i·11-s − i·13-s + 1.41·17-s + 19-s − 1.41·23-s − 1.41i·29-s − 31-s + i·43-s + 1.41·47-s − 1.41i·59-s + 61-s + i·67-s + 1.41·77-s + 1.41·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.296424832\)
\(L(\frac12)\) \(\approx\) \(1.296424832\)
\(L(1)\) 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 \( 1 - iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.491319359496130998365278313813, −8.048926040971543152693827590152, −7.41506294075812564212143112229, −6.01053893254590270805515487532, −5.82436061450194777945724221266, −5.17938271585647593745218092500, −3.80522900517092784242027873356, −3.17960553631215887323214806857, −2.30011315684561916291634291834, −0.870213741523515113660782116175, 1.26325894676897359106920697946, 2.17041713219627770168738164403, 3.54246297667616742686760450820, 4.06676293688038397551329429009, 4.97557605076942760042474953897, 5.71893088329926182108478285573, 6.81990408756708762658555072224, 7.36749747949099579525022890121, 7.73590712605350375653661837162, 8.925234705519217595961433380630

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