Properties

Label 2-252-252.191-c1-0-6
Degree $2$
Conductor $252$
Sign $0.917 + 0.396i$
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  + (−0.834 − 1.14i)2-s + (−1.27 − 1.17i)3-s + (−0.608 + 1.90i)4-s + 1.08i·5-s + (−0.274 + 2.43i)6-s + (2.19 + 1.48i)7-s + (2.68 − 0.894i)8-s + (0.252 + 2.98i)9-s + (1.24 − 0.906i)10-s − 1.50·11-s + (3.00 − 1.71i)12-s + (1.59 + 2.75i)13-s + (−0.131 − 3.73i)14-s + (1.27 − 1.38i)15-s + (−3.26 − 2.31i)16-s + (5.40 − 3.11i)17-s + ⋯
L(s)  = 1  + (−0.589 − 0.807i)2-s + (−0.736 − 0.676i)3-s + (−0.304 + 0.952i)4-s + 0.486i·5-s + (−0.112 + 0.993i)6-s + (0.827 + 0.561i)7-s + (0.948 − 0.316i)8-s + (0.0842 + 0.996i)9-s + (0.392 − 0.286i)10-s − 0.453·11-s + (0.868 − 0.495i)12-s + (0.441 + 0.764i)13-s + (−0.0352 − 0.999i)14-s + (0.328 − 0.357i)15-s + (−0.815 − 0.579i)16-s + (1.31 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.917 + 0.396i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.917 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763977 - 0.157994i\)
\(L(\frac12)\) \(\approx\) \(0.763977 - 0.157994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.834 + 1.14i)T \)
3 \( 1 + (1.27 + 1.17i)T \)
7 \( 1 + (-2.19 - 1.48i)T \)
good5 \( 1 - 1.08iT - 5T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
13 \( 1 + (-1.59 - 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.40 + 3.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.29 - 1.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.25T + 23T^{2} \)
29 \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.708 - 0.409i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.35 - 7.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.57 + 3.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.95 - 2.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.81 - 11.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.686 - 0.396i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.19 - 2.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.96 + 4.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + (4.02 + 6.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.4 + 7.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.40 - 4.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.90 + 3.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.69 + 11.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.91182094396174307424930522476, −11.12010917815560637819381963471, −10.37221776395246408617370579252, −9.177759920787238684037898804100, −7.960229786283515373755207826178, −7.34632539848247209666185395210, −5.90212902389155998287055048283, −4.66024435485787807997846645770, −2.83773662751352612475954678180, −1.44394212634052972460503980318, 0.996054677476280045566641163843, 4.00372036742903193273155742171, 5.22711048418530754863653520042, 5.77768446959949301850223439307, 7.27858780333681293958108717699, 8.172418132469607037302571201402, 9.164147229018219689677445006886, 10.38744076040722614428593022264, 10.66022042040783672441125793718, 11.92250241708472206532557515385

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