Properties

Label 2-252-63.38-c1-0-2
Degree $2$
Conductor $252$
Sign $0.534 - 0.845i$
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  + (1.70 − 0.276i)3-s + (−1.95 + 3.39i)5-s + (0.554 + 2.58i)7-s + (2.84 − 0.943i)9-s + (−3.19 + 1.84i)11-s + (0.480 − 0.277i)13-s + (−2.41 + 6.33i)15-s + (2.91 − 5.05i)17-s + (4.62 − 2.66i)19-s + (1.66 + 4.27i)21-s + (−1.96 − 1.13i)23-s + (−5.16 − 8.94i)25-s + (4.60 − 2.40i)27-s + (3.53 + 2.04i)29-s − 8.08i·31-s + ⋯
L(s)  = 1  + (0.987 − 0.159i)3-s + (−0.875 + 1.51i)5-s + (0.209 + 0.977i)7-s + (0.949 − 0.314i)9-s + (−0.964 + 0.556i)11-s + (0.133 − 0.0769i)13-s + (−0.622 + 1.63i)15-s + (0.707 − 1.22i)17-s + (1.06 − 0.612i)19-s + (0.362 + 0.931i)21-s + (−0.410 − 0.237i)23-s + (−1.03 − 1.78i)25-s + (0.886 − 0.461i)27-s + (0.656 + 0.379i)29-s − 1.45i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.534 - 0.845i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.534 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30733 + 0.720104i\)
\(L(\frac12)\) \(\approx\) \(1.30733 + 0.720104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.276i)T \)
7 \( 1 + (-0.554 - 2.58i)T \)
good5 \( 1 + (1.95 - 3.39i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.19 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.480 + 0.277i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.91 + 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.66i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.96 + 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.53 - 2.04i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.08iT - 31T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.59 - 6.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.754 - 1.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (0.0415 + 0.0239i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 + 6.96iT - 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 + 2.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.94T + 79T^{2} \)
83 \( 1 + (-3.84 + 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.71 + 4.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.9 - 8.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.03765846330139907208301662776, −11.38576488275316716396120749872, −10.17077596143290477192363932981, −9.375440347850102898750596816185, −7.888937733984855309474137368537, −7.63082557880754292834672178187, −6.43502961673184269755313692465, −4.77748328010982781055938194624, −3.13993338521245076960918560470, −2.59402507726558448233202860390, 1.24726301421273891321660982179, 3.48034000625143063113917435769, 4.30486915844403127825181429678, 5.45648467582324526211791630716, 7.49247625667552959276086660565, 8.042271998600915781548971060093, 8.718362064676325809018202857343, 9.908395760842225515753944587643, 10.79724628406217840312993843942, 12.15608309913364908892204358898

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