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Properties

Label 2-252-9.4-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.0832 - 0.996i$
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  + (0.796 + 1.53i)3-s + (−1.02 + 1.77i)5-s + (−0.5 − 0.866i)7-s + (−1.73 + 2.45i)9-s + (2.52 + 4.37i)11-s + (0.5 − 0.866i)13-s + (−3.55 − 0.162i)15-s + 0.273·17-s − 5.38·19-s + (0.933 − 1.45i)21-s + (2.66 − 4.61i)23-s + (0.390 + 0.676i)25-s + (−5.14 − 0.708i)27-s + (4.16 + 7.21i)29-s + (5.08 − 8.80i)31-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)3-s + (−0.459 + 0.795i)5-s + (−0.188 − 0.327i)7-s + (−0.576 + 0.816i)9-s + (0.761 + 1.31i)11-s + (0.138 − 0.240i)13-s + (−0.917 − 0.0418i)15-s + 0.0662·17-s − 1.23·19-s + (0.203 − 0.318i)21-s + (0.555 − 0.962i)23-s + (0.0780 + 0.135i)25-s + (−0.990 − 0.136i)27-s + (0.773 + 1.33i)29-s + (0.912 − 1.58i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0832 - 0.996i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.0832 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867887 + 0.943451i\)
\(L(\frac12)\) \(\approx\) \(0.867887 + 0.943451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.796 - 1.53i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.02 - 1.77i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.52 - 4.37i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.273T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + (-2.66 + 4.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.16 - 7.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.08 + 8.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.16T + 37T^{2} \)
41 \( 1 + (-2.52 + 4.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.30 + 3.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.690 + 1.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.43T + 53T^{2} \)
59 \( 1 + (0.890 - 1.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.390 + 0.676i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.19 + 7.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.78T + 71T^{2} \)
73 \( 1 - 9.38T + 73T^{2} \)
79 \( 1 + (6.47 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.86 - 4.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (-1.10 - 1.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.22196548863588868497500324697, −11.05349493708087182387082518610, −10.41929137928389158499989517274, −9.535085168137362020623584938653, −8.507725440296463741338428054880, −7.36730083087809309378885372718, −6.41679944726593406557584588540, −4.68399406015828843288797991777, −3.86315390451844595562142974679, −2.54449787003725554882635162276, 1.06761587383282006321810434411, 2.92378284490732128024108184440, 4.27846002097281827006952561126, 5.91962011969041426407445239974, 6.72762418603509454033685444281, 8.241324239741766782229839783744, 8.534774097398416265791193832353, 9.563441597851516902758463293679, 11.20528713869799952842530977468, 11.90170491411507061163486125505

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