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Properties

Label 2-252-252.115-c1-0-41
Degree $2$
Conductor $252$
Sign $-0.331 + 0.943i$
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  + (1.40 − 0.154i)2-s + (−0.868 − 1.49i)3-s + (1.95 − 0.434i)4-s + (−2.69 − 1.55i)5-s + (−1.45 − 1.97i)6-s + (−2.30 − 1.29i)7-s + (2.67 − 0.913i)8-s + (−1.49 + 2.60i)9-s + (−4.02 − 1.76i)10-s + (1.94 − 1.12i)11-s + (−2.34 − 2.54i)12-s + (3.87 − 2.23i)13-s + (−3.44 − 1.46i)14-s + (0.00952 + 5.38i)15-s + (3.62 − 1.69i)16-s + (−2.92 − 1.68i)17-s + ⋯
L(s)  = 1  + (0.993 − 0.109i)2-s + (−0.501 − 0.865i)3-s + (0.976 − 0.217i)4-s + (−1.20 − 0.695i)5-s + (−0.593 − 0.805i)6-s + (−0.871 − 0.491i)7-s + (0.946 − 0.322i)8-s + (−0.496 + 0.867i)9-s + (−1.27 − 0.559i)10-s + (0.586 − 0.338i)11-s + (−0.677 − 0.735i)12-s + (1.07 − 0.620i)13-s + (−0.919 − 0.392i)14-s + (0.00245 + 1.39i)15-s + (0.905 − 0.424i)16-s + (−0.708 − 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.331 + 0.943i$
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: \((2,\ 252,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875614 - 1.23625i\)
\(L(\frac12)\) \(\approx\) \(0.875614 - 1.23625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.154i)T \)
3 \( 1 + (0.868 + 1.49i)T \)
7 \( 1 + (2.30 + 1.29i)T \)
good5 \( 1 + (2.69 + 1.55i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.94 + 1.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.87 + 2.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.92 + 1.68i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.28 - 5.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.746 - 0.430i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.897 + 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 + (1.93 + 3.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.06 + 0.613i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.846 + 0.488i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (0.992 - 1.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.17T + 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 + (-3.89 - 2.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (-1.48 + 2.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.1 - 7.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.86 + 1.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.86620937744943349142992124895, −11.34633287323116419472644555277, −10.22545919040724756206662967246, −8.477893042361239095724959832897, −7.56701437354353634382480818562, −6.57967686971834811348073577241, −5.67560133335252890364087094159, −4.28191039063064328192467149913, −3.27256644258695729083757461409, −1.01643392534177313756572046822, 3.08545143416675697064530057971, 3.86773343570731248367252890416, 4.88571484082006944601248225487, 6.45302631068532089766646823674, 6.75610026984653573785488205887, 8.419996348080189167050402410276, 9.620202642605577373632059163848, 10.91339714914263715177351372182, 11.44768020294496411612493686285, 12.07605785304833518423427174286

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