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Properties

Label 2-252-252.115-c1-0-4
Degree $2$
Conductor $252$
Sign $-0.163 - 0.986i$
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  + (−1.33 − 0.477i)2-s + (−1.43 + 0.964i)3-s + (1.54 + 1.27i)4-s + (3.01 + 1.74i)5-s + (2.37 − 0.596i)6-s + (−1.90 + 1.83i)7-s + (−1.44 − 2.43i)8-s + (1.13 − 2.77i)9-s + (−3.18 − 3.75i)10-s + (1.71 − 0.987i)11-s + (−3.44 − 0.341i)12-s + (−5.52 + 3.18i)13-s + (3.41 − 1.53i)14-s + (−6.01 + 0.404i)15-s + (0.762 + 3.92i)16-s + (1.04 + 0.600i)17-s + ⋯
L(s)  = 1  + (−0.941 − 0.337i)2-s + (−0.830 + 0.556i)3-s + (0.771 + 0.636i)4-s + (1.34 + 0.778i)5-s + (0.969 − 0.243i)6-s + (−0.719 + 0.694i)7-s + (−0.511 − 0.859i)8-s + (0.379 − 0.925i)9-s + (−1.00 − 1.18i)10-s + (0.515 − 0.297i)11-s + (−0.995 − 0.0985i)12-s + (−1.53 + 0.884i)13-s + (0.912 − 0.410i)14-s + (−1.55 + 0.104i)15-s + (0.190 + 0.981i)16-s + (0.252 + 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.411157 + 0.484977i\)
\(L(\frac12)\) \(\approx\) \(0.411157 + 0.484977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.477i)T \)
3 \( 1 + (1.43 - 0.964i)T \)
7 \( 1 + (1.90 - 1.83i)T \)
good5 \( 1 + (-3.01 - 1.74i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.71 + 0.987i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.52 - 3.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.04 - 0.600i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.25 - 3.90i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.511 + 0.295i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.00 - 3.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 + (-0.506 - 0.877i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.54 - 2.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.66 - 1.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.349T + 47T^{2} \)
53 \( 1 + (-3.01 + 5.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + 3.95iT - 61T^{2} \)
67 \( 1 + 4.89iT - 67T^{2} \)
71 \( 1 - 1.47iT - 71T^{2} \)
73 \( 1 + (-9.61 - 5.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.642iT - 79T^{2} \)
83 \( 1 + (-1.00 + 1.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.27 - 1.88i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.8 - 8.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.03492820183865628277717462697, −11.20466915569665835281269092309, −9.981280506640246755924846474655, −9.824266042996622519316179313454, −8.980079469993963548352769877819, −7.10301958288524658361235787300, −6.38363893552958798549542002863, −5.46195003205099808188161422232, −3.43809692499560736167624680622, −2.04316421245804107985309708840, 0.73394480558213157928953546239, 2.25356409313121203374089657855, 5.05402584080223534559355987293, 5.79968833346297142083355965830, 6.89968549756370142272281055377, 7.58177799018587305164858427979, 9.142359669731169022785041144446, 9.869332109064997900155797761655, 10.44239387728864714828534708580, 11.76624381918663295020710035419

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