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Properties

Label 2-252-252.103-c1-0-3
Degree $2$
Conductor $252$
Sign $0.694 - 0.719i$
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  + (−0.379 − 1.36i)2-s + (−1.39 − 1.02i)3-s + (−1.71 + 1.03i)4-s + (−2.34 + 1.35i)5-s + (−0.862 + 2.29i)6-s + (−1.09 − 2.40i)7-s + (2.06 + 1.93i)8-s + (0.906 + 2.85i)9-s + (2.73 + 2.68i)10-s + (3.92 + 2.26i)11-s + (3.45 + 0.304i)12-s + (1.85 + 1.06i)13-s + (−2.86 + 2.41i)14-s + (4.66 + 0.507i)15-s + (1.85 − 3.54i)16-s + (−5.38 + 3.10i)17-s + ⋯
L(s)  = 1  + (−0.268 − 0.963i)2-s + (−0.806 − 0.590i)3-s + (−0.855 + 0.517i)4-s + (−1.04 + 0.606i)5-s + (−0.352 + 0.935i)6-s + (−0.415 − 0.909i)7-s + (0.728 + 0.685i)8-s + (0.302 + 0.953i)9-s + (0.865 + 0.848i)10-s + (1.18 + 0.683i)11-s + (0.996 + 0.0879i)12-s + (0.513 + 0.296i)13-s + (−0.764 + 0.644i)14-s + (1.20 + 0.131i)15-s + (0.464 − 0.885i)16-s + (−1.30 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.694 - 0.719i$
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: \((2,\ 252,\ (\ :1/2),\ 0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299717 + 0.127384i\)
\(L(\frac12)\) \(\approx\) \(0.299717 + 0.127384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.379 + 1.36i)T \)
3 \( 1 + (1.39 + 1.02i)T \)
7 \( 1 + (1.09 + 2.40i)T \)
good5 \( 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.92 - 2.26i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.85 - 1.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.38 - 3.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.634 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.14 - 4.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 5.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.80T + 31T^{2} \)
37 \( 1 + (-1.47 + 2.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.01 - 2.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.36 - 0.790i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.56T + 47T^{2} \)
53 \( 1 + (1.70 + 2.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.01T + 59T^{2} \)
61 \( 1 - 4.50iT - 61T^{2} \)
67 \( 1 - 8.13iT - 67T^{2} \)
71 \( 1 + 7.04iT - 71T^{2} \)
73 \( 1 + (-9.12 + 5.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.64iT - 79T^{2} \)
83 \( 1 + (0.131 + 0.228i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.21 - 1.27i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.4 - 7.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.95795716093275543950293842309, −11.22519636203137449604543973464, −10.65866467749608570823386625423, −9.562392791765497499739731905582, −8.181952526821279577702682387011, −7.23631202742584537954155168258, −6.38968890440655763856685379111, −4.37688726855547223650568809222, −3.71668559067629419095696466169, −1.63776146159116201511150936025, 0.32658094592776460020423925778, 3.88200922259768655311838870151, 4.70402486707731413313738872695, 6.01538469466405223724827466551, 6.60419980360826622402272854294, 8.208745690070193850702583971767, 8.902141986165179573044792677106, 9.697408238176408888763892115380, 11.08150647249750685225648276944, 11.85059976334665595908492430849

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