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Properties

Label 2-468-117.103-c1-0-9
Degree $2$
Conductor $468$
Sign $0.998 + 0.0483i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (1.61 − 0.613i)3-s + (2.41 + 1.39i)5-s + (−1.00 + 0.582i)7-s + (2.24 − 1.98i)9-s + (0.716 − 0.413i)11-s + (−0.360 − 3.58i)13-s + (4.76 + 0.775i)15-s + 0.0570·17-s + 8.44i·19-s + (−1.27 + 1.56i)21-s + (−1.18 + 2.04i)23-s + (1.37 + 2.38i)25-s + (2.42 − 4.59i)27-s + (−2.67 − 4.62i)29-s + (0.351 + 0.203i)31-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)3-s + (1.07 + 0.622i)5-s + (−0.381 + 0.219i)7-s + (0.749 − 0.662i)9-s + (0.216 − 0.124i)11-s + (−0.0999 − 0.994i)13-s + (1.22 + 0.200i)15-s + 0.0138·17-s + 1.93i·19-s + (−0.278 + 0.340i)21-s + (−0.246 + 0.426i)23-s + (0.275 + 0.477i)25-s + (0.465 − 0.884i)27-s + (−0.495 − 0.858i)29-s + (0.0631 + 0.0364i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.998 + 0.0483i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.998 + 0.0483i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18448 - 0.0528602i\)
\(L(\frac12)\) \(\approx\) \(2.18448 - 0.0528602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.613i)T \)
13 \( 1 + (0.360 + 3.58i)T \)
good5 \( 1 + (-2.41 - 1.39i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.00 - 0.582i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.716 + 0.413i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.0570T + 17T^{2} \)
19 \( 1 - 8.44iT - 19T^{2} \)
23 \( 1 + (1.18 - 2.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.351 - 0.203i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.42iT - 37T^{2} \)
41 \( 1 + (7.53 + 4.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.13 - 8.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.56 + 4.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.12T + 53T^{2} \)
59 \( 1 + (9.68 + 5.58i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.225 + 0.389i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.02 + 4.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.15iT - 71T^{2} \)
73 \( 1 - 8.45iT - 73T^{2} \)
79 \( 1 + (-4.40 - 7.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.12 - 5.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.24iT - 89T^{2} \)
97 \( 1 + (0.0761 - 0.0439i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74229733222836604691869210240, −9.916275180222177799087275185927, −9.433724486765141065213410091284, −8.239137833652171473148406397837, −7.48800506977586971105714563156, −6.28232185684019966418834903519, −5.69842058641874235821404837896, −3.83365232297210965098798147332, −2.83940541225189305339744554406, −1.74013833100634219985121928711, 1.68328858476747133062995784067, 2.85971292154606428507946727050, 4.28487874688746470910708100954, 5.12220945402584628907138259999, 6.50790416895742257676597288799, 7.34528796498294039411684778631, 8.825362250990704413009682762495, 9.100906041866586187907808264551, 9.902317648063267010068581383123, 10.78061616176067952404677914343

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