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Properties

Label 2-252-252.103-c1-0-29
Degree $2$
Conductor $252$
Sign $0.399 + 0.916i$
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 − 0.140i)2-s + (1.46 − 0.920i)3-s + (1.96 + 0.396i)4-s + (0.259 − 0.149i)5-s + (−2.19 + 1.08i)6-s + (0.0899 − 2.64i)7-s + (−2.70 − 0.833i)8-s + (1.30 − 2.70i)9-s + (−0.385 + 0.174i)10-s + (0.0516 + 0.0297i)11-s + (3.24 − 1.22i)12-s + (1.63 + 0.942i)13-s + (−0.498 + 3.70i)14-s + (0.242 − 0.458i)15-s + (3.68 + 1.55i)16-s + (−5.35 + 3.08i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0995i)2-s + (0.847 − 0.531i)3-s + (0.980 + 0.198i)4-s + (0.115 − 0.0669i)5-s + (−0.895 + 0.444i)6-s + (0.0340 − 0.999i)7-s + (−0.955 − 0.294i)8-s + (0.435 − 0.900i)9-s + (−0.122 + 0.0550i)10-s + (0.0155 + 0.00898i)11-s + (0.935 − 0.352i)12-s + (0.452 + 0.261i)13-s + (−0.133 + 0.991i)14-s + (0.0626 − 0.118i)15-s + (0.921 + 0.388i)16-s + (−1.29 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.399 + 0.916i$
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: \((2,\ 252,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915153 - 0.599389i\)
\(L(\frac12)\) \(\approx\) \(0.915153 - 0.599389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.140i)T \)
3 \( 1 + (-1.46 + 0.920i)T \)
7 \( 1 + (-0.0899 + 2.64i)T \)
good5 \( 1 + (-0.259 + 0.149i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0516 - 0.0297i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.63 - 0.942i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.35 - 3.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 3.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.99 + 2.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.70 + 2.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.28T + 31T^{2} \)
37 \( 1 + (-1.19 + 2.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.16 - 5.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.50 - 4.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.91T + 47T^{2} \)
53 \( 1 + (-3.44 - 5.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.893T + 59T^{2} \)
61 \( 1 - 7.83iT - 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 - 1.25iT - 71T^{2} \)
73 \( 1 + (6.43 - 3.71i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.54iT - 79T^{2} \)
83 \( 1 + (-4.61 - 7.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.82 - 5.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.40 - 1.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.60367117358181875655194335862, −10.84388489020187991800398319584, −9.725836450810991930058159999711, −8.933052057107837522596242013688, −8.055019822071350193542394799055, −7.10315239100394833599454151380, −6.37995737486530176644235424249, −4.15414436195443365484060605891, −2.73357954589789234129838564054, −1.22229705848000149651125300466, 2.07944654209005610439129782147, 3.22913296182153481164940316672, 5.08316636632428157941920240633, 6.37896866265026524823774594073, 7.64386794283275204816009138711, 8.584770310624664187941363473459, 9.193795635121629974677774368013, 10.03815351716021850457869967438, 11.04405752025737991425798953626, 11.93174999002056919621814593994

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