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Properties

Label 2-252-252.103-c1-0-24
Degree $2$
Conductor $252$
Sign $0.923 + 0.384i$
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  + (1.06 − 0.928i)2-s + (−0.187 + 1.72i)3-s + (0.275 − 1.98i)4-s + (2.33 − 1.34i)5-s + (1.39 + 2.01i)6-s + (1.11 + 2.39i)7-s + (−1.54 − 2.36i)8-s + (−2.92 − 0.645i)9-s + (1.23 − 3.60i)10-s + (−1.22 − 0.706i)11-s + (3.35 + 0.845i)12-s + (2.46 + 1.42i)13-s + (3.41 + 1.51i)14-s + (1.88 + 4.26i)15-s + (−3.84 − 1.09i)16-s + (−1.23 + 0.712i)17-s + ⋯
L(s)  = 1  + (0.754 − 0.656i)2-s + (−0.108 + 0.994i)3-s + (0.137 − 0.990i)4-s + (1.04 − 0.602i)5-s + (0.571 + 0.820i)6-s + (0.422 + 0.906i)7-s + (−0.546 − 0.837i)8-s + (−0.976 − 0.215i)9-s + (0.391 − 1.13i)10-s + (−0.369 − 0.213i)11-s + (0.969 + 0.244i)12-s + (0.683 + 0.394i)13-s + (0.914 + 0.405i)14-s + (0.485 + 1.10i)15-s + (−0.962 − 0.272i)16-s + (−0.299 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96552 - 0.392497i\)
\(L(\frac12)\) \(\approx\) \(1.96552 - 0.392497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.928i)T \)
3 \( 1 + (0.187 - 1.72i)T \)
7 \( 1 + (-1.11 - 2.39i)T \)
good5 \( 1 + (-2.33 + 1.34i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.22 + 0.706i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.23 - 0.712i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.67 + 0.967i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.423 - 0.733i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.87T + 31T^{2} \)
37 \( 1 + (5.49 - 9.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.99 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.96 - 3.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + (6.58 + 11.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 + 2.03iT - 61T^{2} \)
67 \( 1 - 8.74iT - 67T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 + (-7.16 + 4.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 + (-2.85 - 4.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.75 - 3.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.1 + 7.57i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.85080617610590575933829206907, −11.10548342227404216880962089461, −10.20465094194042664005658649623, −9.210574740763359643980023999373, −8.705632308223816812362262392652, −6.37787809225468958479912454978, −5.38446002777403559562764868409, −4.90566863666990757999671506605, −3.37067205783138889242455249272, −1.93665883195229071762941112635, 2.01021835965664099681674113601, 3.49877376018804437401932814624, 5.25005732206147397672394555994, 6.07134621644968291931283276736, 7.07302833167671499371427197732, 7.69985650974069174340949159666, 8.909763605008216102997218208259, 10.49158849978284576812319136690, 11.24146331126776395466538459975, 12.47251281300363335408145228029

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