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Properties

Label 2-252-252.139-c1-0-23
Degree $2$
Conductor $252$
Sign $0.671 + 0.740i$
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  + (−1.28 − 0.582i)2-s + (1.60 − 0.649i)3-s + (1.32 + 1.50i)4-s + (1.35 + 0.782i)5-s + (−2.44 − 0.0983i)6-s + (−0.314 − 2.62i)7-s + (−0.825 − 2.70i)8-s + (2.15 − 2.08i)9-s + (−1.28 − 1.79i)10-s + (−2.65 + 1.53i)11-s + (3.09 + 1.55i)12-s + (4.17 + 2.41i)13-s + (−1.12 + 3.56i)14-s + (2.68 + 0.375i)15-s + (−0.512 + 3.96i)16-s − 0.603i·17-s + ⋯
L(s)  = 1  + (−0.911 − 0.412i)2-s + (0.926 − 0.375i)3-s + (0.660 + 0.751i)4-s + (0.605 + 0.349i)5-s + (−0.999 − 0.0401i)6-s + (−0.118 − 0.992i)7-s + (−0.291 − 0.956i)8-s + (0.718 − 0.695i)9-s + (−0.407 − 0.568i)10-s + (−0.800 + 0.462i)11-s + (0.893 + 0.448i)12-s + (1.15 + 0.669i)13-s + (−0.301 + 0.953i)14-s + (0.692 + 0.0968i)15-s + (−0.128 + 0.991i)16-s − 0.146i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12587 - 0.498724i\)
\(L(\frac12)\) \(\approx\) \(1.12587 - 0.498724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.582i)T \)
3 \( 1 + (-1.60 + 0.649i)T \)
7 \( 1 + (0.314 + 2.62i)T \)
good5 \( 1 + (-1.35 - 0.782i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.65 - 1.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.17 - 2.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.603iT - 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 + (7.81 + 4.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.38 - 5.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.860 - 1.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.45T + 37T^{2} \)
41 \( 1 + (-2.08 - 1.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.473 - 0.273i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.430T + 53T^{2} \)
59 \( 1 + (2.05 - 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.39 - 0.803i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 - 1.88iT - 73T^{2} \)
79 \( 1 + (6.41 - 3.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.801 - 1.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
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

−11.87224197435378703574549709523, −10.51659689948130951324908143706, −10.08298000688906233964316693041, −9.064085402175416238667872632696, −8.071794624665271402859824281332, −7.22608828509522821584223256762, −6.34998743828508819411827571786, −4.05526379484728759278828220434, −2.84270585867154652078087530159, −1.49825924219297854182222319878, 1.86501656725652825637899280493, 3.22340406921888011178510560817, 5.35728440592951607781280620594, 6.00101596950553430953107402599, 7.76036056803122616729253086804, 8.307327463509568516205054064413, 9.275281116957936535365338718139, 9.869127665027684838290743870103, 10.85406822553215721234464180133, 12.05115651079964302816769525149

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