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Properties

Label 2-252-63.5-c1-0-3
Degree $2$
Conductor $252$
Sign $0.708 - 0.705i$
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  + (1.68 + 0.418i)3-s + (1.37 + 2.37i)5-s + (−2.60 + 0.463i)7-s + (2.65 + 1.40i)9-s + (−0.362 − 0.209i)11-s + (1.32 + 0.765i)13-s + (1.31 + 4.56i)15-s + (−1.95 − 3.38i)17-s + (−5.11 − 2.95i)19-s + (−4.57 − 0.309i)21-s + (7.72 − 4.46i)23-s + (−1.26 + 2.18i)25-s + (3.86 + 3.47i)27-s + (6.00 − 3.46i)29-s + 3.52i·31-s + ⋯
L(s)  = 1  + (0.970 + 0.241i)3-s + (0.613 + 1.06i)5-s + (−0.984 + 0.175i)7-s + (0.883 + 0.468i)9-s + (−0.109 − 0.0630i)11-s + (0.367 + 0.212i)13-s + (0.338 + 1.17i)15-s + (−0.473 − 0.820i)17-s + (−1.17 − 0.678i)19-s + (−0.997 − 0.0674i)21-s + (1.61 − 0.930i)23-s + (−0.252 + 0.437i)25-s + (0.744 + 0.667i)27-s + (1.11 − 0.643i)29-s + 0.633i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.708 - 0.705i$
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: \((2,\ 252,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56407 + 0.646336i\)
\(L(\frac12)\) \(\approx\) \(1.56407 + 0.646336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.418i)T \)
7 \( 1 + (2.60 - 0.463i)T \)
good5 \( 1 + (-1.37 - 2.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.362 + 0.209i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.32 - 0.765i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.72 + 4.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.00 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.52iT - 31T^{2} \)
37 \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.06 + 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.77 + 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.77T + 47T^{2} \)
53 \( 1 + (3.39 - 1.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.05T + 59T^{2} \)
61 \( 1 + 1.86iT - 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.867T + 79T^{2} \)
83 \( 1 + (-3.45 - 5.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.200 + 0.115i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.33428560675635372238845333900, −10.80845243902836479657770859197, −10.28189000953891910447471013815, −9.237429504493670841909767062477, −8.540298379985326596474816206968, −6.89187352684808854844654052275, −6.56100084170261127547120122185, −4.73996670756605850971546548876, −3.18945251692596944068788348385, −2.46938430979405413340116735714, 1.53608820074980497259687498107, 3.15856091012414130330540439182, 4.43488745180310512407192383180, 5.92001493029038132943217514673, 6.96026953200161593465051997867, 8.295765990573993938024289262390, 8.983715532723879585332202427620, 9.746961312844519578626616012882, 10.74833618418184855751385262731, 12.45145428201346159578995786956

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