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Properties

Label 2-3636-3636.1615-c0-0-10
Degree $2$
Conductor $3636$
Sign $0.456 + 0.889i$
Analytic cond. $1.81460$
Root an. cond. $1.34707$
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  + (−0.5 − 0.866i)2-s + (0.623 + 0.781i)3-s + (−0.499 + 0.866i)4-s + (0.900 − 1.56i)5-s + (0.365 − 0.930i)6-s + (0.733 + 1.26i)7-s + 0.999·8-s + (−0.222 + 0.974i)9-s − 1.80·10-s + (−0.826 − 1.43i)11-s + (−0.988 + 0.149i)12-s + (0.733 − 1.26i)13-s + (0.733 − 1.26i)14-s + (1.78 − 0.268i)15-s + (−0.5 − 0.866i)16-s + 1.24·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.623 + 0.781i)3-s + (−0.499 + 0.866i)4-s + (0.900 − 1.56i)5-s + (0.365 − 0.930i)6-s + (0.733 + 1.26i)7-s + 0.999·8-s + (−0.222 + 0.974i)9-s − 1.80·10-s + (−0.826 − 1.43i)11-s + (−0.988 + 0.149i)12-s + (0.733 − 1.26i)13-s + (0.733 − 1.26i)14-s + (1.78 − 0.268i)15-s + (−0.5 − 0.866i)16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3636\)    =    \(2^{2} \cdot 3^{2} \cdot 101\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(1.81460\)
Root analytic conductor: \(1.34707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3636} (1615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3636,\ (\ :0),\ 0.456 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504579541\)
\(L(\frac12)\) \(\approx\) \(1.504579541\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.623 - 0.781i)T \)
101 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.24T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.97T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.704026562219558171118943934313, −8.206620084204558411094448451714, −7.945842560881285585839545207915, −5.77436657190382665497636056590, −5.35831772657141561394209058330, −4.98606794244062882215602937222, −3.67594709030476769315585582738, −2.95141412664299807187757223105, −2.07040747119573435886650062087, −1.06264093233085582173357294919, 1.55742941825531324251839102472, 1.97556127701846412235390799138, 3.31248989267222172243143571285, 4.28036219950461918223782061325, 5.34634381417353027684233195595, 6.28539997989427414442282056669, 6.91131722563178595346225139333, 7.34042216914415535732883877224, 7.71453332270622325380261287489, 8.693163444363506334785803690355

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