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Properties

Label 2-252-252.115-c1-0-37
Degree $2$
Conductor $252$
Sign $-0.959 + 0.280i$
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 + (−1.92 − 1.51i)6-s + (1.09 − 2.40i)7-s + (2.06 + 1.93i)8-s + (0.906 − 2.85i)9-s + (−0.954 + 3.71i)10-s + (−3.92 + 2.26i)11-s + (−1.33 + 3.19i)12-s + (1.85 − 1.06i)13-s + (−3.69 − 0.584i)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 + (−0.785 − 0.618i)6-s + (0.415 − 0.909i)7-s + (0.728 + 0.685i)8-s + (0.302 − 0.953i)9-s + (−0.301 + 1.17i)10-s + (−1.18 + 0.683i)11-s + (−0.384 + 0.923i)12-s + (0.513 − 0.296i)13-s + (−0.987 − 0.156i)14-s + (−1.20 + 0.131i)15-s + (0.464 − 0.885i)16-s + (−1.30 − 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.959 + 0.280i$
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: \((2,\ 252,\ (\ :1/2),\ -0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143003 - 0.999967i\)
\(L(\frac12)\) \(\approx\) \(0.143003 - 0.999967i\)
\(L(\frac{3}{2})\) 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

−11.61689581861438100900101099759, −10.80859714767224120254299797859, −9.649984275630499564787669181806, −8.610388977521644670858478561101, −7.85761858772132010808669839923, −7.23035124419827827753523419624, −4.81683901697589215954295277896, −3.95382607441535345602147003669, −2.59587731692122258725141133718, −0.847477837755094493597091479790, 2.80384680973420203552047745430, 4.21355529865260302783274578447, 5.25740344580690327254604498893, 6.68862849128151770009947062755, 7.82977496659025129534498190670, 8.525054183105801526306002593748, 9.095655383664236263164458503075, 10.72267682494709129110977208328, 11.00357450391934923890848837445, 12.75668420689681280783587678422

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