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Properties

Label 2-252-36.23-c1-0-29
Degree $2$
Conductor $252$
Sign $0.265 + 0.964i$
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  + (0.791 − 1.17i)2-s + (1.72 − 0.202i)3-s + (−0.748 − 1.85i)4-s + (0.795 + 0.459i)5-s + (1.12 − 2.17i)6-s + (−0.866 + 0.5i)7-s + (−2.76 − 0.589i)8-s + (2.91 − 0.697i)9-s + (1.16 − 0.569i)10-s + (−0.582 − 1.00i)11-s + (−1.66 − 3.03i)12-s + (0.0273 − 0.0472i)13-s + (−0.0989 + 1.41i)14-s + (1.46 + 0.628i)15-s + (−2.87 + 2.77i)16-s + 3.29i·17-s + ⋯
L(s)  = 1  + (0.559 − 0.828i)2-s + (0.993 − 0.117i)3-s + (−0.374 − 0.927i)4-s + (0.355 + 0.205i)5-s + (0.458 − 0.888i)6-s + (−0.327 + 0.188i)7-s + (−0.978 − 0.208i)8-s + (0.972 − 0.232i)9-s + (0.369 − 0.180i)10-s + (−0.175 − 0.304i)11-s + (−0.480 − 0.877i)12-s + (0.00757 − 0.0131i)13-s + (−0.0264 + 0.377i)14-s + (0.377 + 0.162i)15-s + (−0.719 + 0.694i)16-s + 0.799i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67757 - 1.27759i\)
\(L(\frac12)\) \(\approx\) \(1.67757 - 1.27759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.791 + 1.17i)T \)
3 \( 1 + (-1.72 + 0.202i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.795 - 0.459i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.582 + 1.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0273 + 0.0472i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.29iT - 17T^{2} \)
19 \( 1 - 0.455iT - 19T^{2} \)
23 \( 1 + (1.77 - 3.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.58 - 4.37i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.03 - 4.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.20T + 37T^{2} \)
41 \( 1 + (5.27 + 3.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.64 + 3.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.82 + 4.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + (0.744 - 1.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.35 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.83 - 3.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + (10.9 - 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.56 + 11.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.95iT - 89T^{2} \)
97 \( 1 + (4.26 + 7.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.09609776568252084371694495970, −10.79003665249061663866846451190, −9.997412719808273107858709739768, −9.167340221200673847341513769522, −8.166149601602958886672784804846, −6.69880482999032720361631827727, −5.57945980358297293582196535596, −4.06066902962683143118249977837, −3.05131112354583653059805266984, −1.81994599190756247443850692261, 2.54077016816534316066598412079, 3.85781681965965355380548874193, 4.92722963356745054333827622586, 6.26834760172384571913736194191, 7.38008942582642496152401950784, 8.141491338773419789004085823291, 9.290930819363535901156930559883, 9.871717914823418767403341376505, 11.50560738578089501857032886527, 12.73023457679230792489329312222

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