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Properties

Label 2-252-63.20-c1-0-6
Degree $2$
Conductor $252$
Sign $-0.907 + 0.419i$
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  + (−1.10 − 1.33i)3-s + (−0.266 − 0.462i)5-s + (−1.89 − 1.84i)7-s + (−0.565 + 2.94i)9-s + (−3.39 − 1.96i)11-s + (0.116 − 0.0674i)13-s + (−0.322 + 0.866i)15-s − 4.32·17-s − 2.22i·19-s + (−0.378 + 4.56i)21-s + (−1.70 + 0.983i)23-s + (2.35 − 4.08i)25-s + (4.55 − 2.49i)27-s + (−5.16 − 2.98i)29-s + (0.800 − 0.462i)31-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)3-s + (−0.119 − 0.206i)5-s + (−0.715 − 0.698i)7-s + (−0.188 + 0.982i)9-s + (−1.02 − 0.591i)11-s + (0.0324 − 0.0187i)13-s + (−0.0832 + 0.223i)15-s − 1.04·17-s − 0.511i·19-s + (−0.0826 + 0.996i)21-s + (−0.355 + 0.205i)23-s + (0.471 − 0.816i)25-s + (0.877 − 0.480i)27-s + (−0.959 − 0.553i)29-s + (0.143 − 0.0829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.907 + 0.419i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115383 - 0.524110i\)
\(L(\frac12)\) \(\approx\) \(0.115383 - 0.524110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.10 + 1.33i)T \)
7 \( 1 + (1.89 + 1.84i)T \)
good5 \( 1 + (0.266 + 0.462i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.39 + 1.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.116 + 0.0674i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.32T + 17T^{2} \)
19 \( 1 + 2.22iT - 19T^{2} \)
23 \( 1 + (1.70 - 0.983i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.16 + 2.98i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.800 + 0.462i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + (-4.59 - 7.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.04 + 5.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + (-1.89 - 3.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.35 - 5.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 + 0.381iT - 73T^{2} \)
79 \( 1 + (4.60 - 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.28 + 2.21i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + (13.6 + 7.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.54110077495807102662036151067, −10.87033947048382495286337894912, −9.901100448202511531125796055945, −8.517136985580541040973052789738, −7.55509899188923398170499198828, −6.61209414527615212498027713374, −5.64257688724316977655888279055, −4.32275247962782484381762768899, −2.56728299692620831787556529294, −0.43494931597950584778927546313, 2.69279369168447787110256183879, 4.11752721307590403493086082995, 5.33527982046298394277078062881, 6.22824871431626828462611705979, 7.42049914546536500659440887121, 8.882086819690559262755564383001, 9.646284801468176414698430209308, 10.60213572522808983412886851089, 11.33720628188989493609259084494, 12.48436572745062518177976282600

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