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Properties

Label 2-252-12.11-c1-0-6
Degree $2$
Conductor $252$
Sign $0.999 + 0.0341i$
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  + (1.35 − 0.405i)2-s + (1.67 − 1.09i)4-s + 3.31i·5-s − i·7-s + (1.81 − 2.16i)8-s + (1.34 + 4.48i)10-s + 4.72·11-s − 4.97·13-s + (−0.405 − 1.35i)14-s + (1.58 − 3.67i)16-s + 0.484i·17-s − 2.29i·19-s + (3.63 + 5.53i)20-s + (6.40 − 1.91i)22-s − 7.97·23-s + ⋯
L(s)  = 1  + (0.958 − 0.286i)2-s + (0.835 − 0.549i)4-s + 1.48i·5-s − 0.377i·7-s + (0.643 − 0.765i)8-s + (0.424 + 1.41i)10-s + 1.42·11-s − 1.38·13-s + (−0.108 − 0.362i)14-s + (0.396 − 0.917i)16-s + 0.117i·17-s − 0.525i·19-s + (0.813 + 1.23i)20-s + (1.36 − 0.408i)22-s − 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0341i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.999 + 0.0341i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16464 - 0.0369601i\)
\(L(\frac12)\) \(\approx\) \(2.16464 - 0.0369601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.405i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 3.31iT - 5T^{2} \)
11 \( 1 - 4.72T + 11T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 - 0.484iT - 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + 7.97T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 + 2.39T + 37T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 - 5.37iT - 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 + 1.00iT - 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 - 3.27iT - 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 - 5.72iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−11.79722324742835936439457124156, −11.41249333527498761606979552432, −10.24244330862356942790848010250, −9.661762965380425434721908562630, −7.65640957591422463077536474229, −6.80782312866686250662816999002, −6.10376163775855954087958455211, −4.49066487021205547009393310048, −3.44830454432889321846769754319, −2.20117831659463861496955943204, 1.89090127530682706920662359393, 3.82376832259150188931083497172, 4.80689709861664701538979324409, 5.69260840525412724197474595860, 6.89094191513584650601634236675, 8.112737833370005258629647941877, 8.996027296171327850303157369217, 10.07881733907071220330901049106, 11.79834332978403068366907688114, 12.12107275433841119703588902733

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