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Properties

Label 2-252-63.5-c1-0-6
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: $\chi_{252} (5, \cdot )$
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

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

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