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Properties

Label 2-252-252.103-c1-0-33
Degree $2$
Conductor $252$
Sign $0.993 + 0.117i$
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  + (1.06 + 0.928i)2-s + (0.187 − 1.72i)3-s + (0.275 + 1.98i)4-s + (2.33 − 1.34i)5-s + (1.79 − 1.66i)6-s + (−1.11 − 2.39i)7-s + (−1.54 + 2.36i)8-s + (−2.92 − 0.645i)9-s + (3.73 + 0.729i)10-s + (1.22 + 0.706i)11-s + (3.46 − 0.102i)12-s + (2.46 + 1.42i)13-s + (1.03 − 3.59i)14-s + (−1.88 − 4.26i)15-s + (−3.84 + 1.09i)16-s + (−1.23 + 0.712i)17-s + ⋯
L(s)  = 1  + (0.754 + 0.656i)2-s + (0.108 − 0.994i)3-s + (0.137 + 0.990i)4-s + (1.04 − 0.602i)5-s + (0.734 − 0.678i)6-s + (−0.422 − 0.906i)7-s + (−0.546 + 0.837i)8-s + (−0.976 − 0.215i)9-s + (1.18 + 0.230i)10-s + (0.369 + 0.213i)11-s + (0.999 − 0.0297i)12-s + (0.683 + 0.394i)13-s + (0.275 − 0.961i)14-s + (−0.485 − 1.10i)15-s + (−0.962 + 0.272i)16-s + (−0.299 + 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.993 + 0.117i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03423 - 0.119949i\)
\(L(\frac12)\) \(\approx\) \(2.03423 - 0.119949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.928i)T \)
3 \( 1 + (-0.187 + 1.72i)T \)
7 \( 1 + (1.11 + 2.39i)T \)
good5 \( 1 + (-2.33 + 1.34i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.22 - 0.706i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.23 - 0.712i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.67 - 0.967i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.423 - 0.733i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
37 \( 1 + (5.49 - 9.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.99 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.96 + 3.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (6.58 + 11.4i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.42T + 59T^{2} \)
61 \( 1 + 2.03iT - 61T^{2} \)
67 \( 1 + 8.74iT - 67T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 + (-7.16 + 4.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (2.85 + 4.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.75 - 3.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.1 + 7.57i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.41965641622598992116863766094, −11.46646707782579546180846249444, −10.01317284783807617074216134539, −8.801081562683130798315187177391, −7.958909860340404972036160745300, −6.60554010775678232678641285859, −6.31095367186358890600852139601, −4.94470540910677377096614909036, −3.50384248555970742887846037085, −1.74246618595803270365014524357, 2.37344937214854406365998413943, 3.28448565903472588982795899711, 4.70074412875138661867333677581, 5.89481305233259113770506042849, 6.37303490283202544879194745479, 8.668553759776942333809745893823, 9.504923171556339067888212867727, 10.24702010123379284843648593600, 11.03231563208110212506418935748, 11.93559885902258542052362746378

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