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Properties

Label 2-252-28.27-c1-0-14
Degree $2$
Conductor $252$
Sign $0.661 + 0.750i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (1.32 − 0.5i)2-s + (1.50 − 1.32i)4-s − 2.64i·7-s + (1.32 − 2.50i)8-s + 4i·11-s + (−1.32 − 3.50i)14-s + (0.500 − 3.96i)16-s + (2 + 5.29i)22-s + 8i·23-s + 5·25-s + (−3.50 − 3.96i)28-s − 10.5·29-s + (−1.32 − 5.50i)32-s − 6·37-s + 5.29i·43-s + (5.29 + 6.00i)44-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + (0.750 − 0.661i)4-s − 0.999i·7-s + (0.467 − 0.883i)8-s + 1.20i·11-s + (−0.353 − 0.935i)14-s + (0.125 − 0.992i)16-s + (0.426 + 1.12i)22-s + 1.66i·23-s + 25-s + (−0.661 − 0.749i)28-s − 1.96·29-s + (−0.233 − 0.972i)32-s − 0.986·37-s + 0.806i·43-s + (0.797 + 0.904i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91635 - 0.865072i\)
\(L(\frac12)\) \(\approx\) \(1.91635 - 0.865072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + 2.64iT \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 16iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.98463587241913786861598698231, −11.08061618263022067579723605664, −10.22011999905926563756067462453, −9.369940871577084845378775056533, −7.51514557866725144703633220496, −6.97417319036628040235627563948, −5.54748321384062432046459152552, −4.46865325691232563849339806941, −3.44127348092956702547536900357, −1.70012766171373320250956563777, 2.43370948788322615610044689888, 3.65096293991075743902802151125, 5.12278994592470895151430590929, 5.93147938552522879250393834542, 6.93731582991745748630645217553, 8.290023436584896162279437184041, 8.947268348943713922360403359754, 10.62040838504365737126129127924, 11.41445173873599080963192176355, 12.34488755835955889693701981728

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