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Properties

Label 2-1338-223.39-c1-0-11
Degree $2$
Conductor $1338$
Sign $0.909 - 0.415i$
Analytic cond. $10.6839$
Root an. cond. $3.26863$
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  + 2-s + (0.5 − 0.866i)3-s + 4-s + (0.887 + 1.53i)5-s + (0.5 − 0.866i)6-s − 3.74·7-s + 8-s + (−0.499 − 0.866i)9-s + (0.887 + 1.53i)10-s + (2.36 + 4.10i)11-s + (0.5 − 0.866i)12-s + 2.39·13-s − 3.74·14-s + 1.77·15-s + 16-s + 4.75·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.396 + 0.687i)5-s + (0.204 − 0.353i)6-s − 1.41·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.280 + 0.485i)10-s + (0.714 + 1.23i)11-s + (0.144 − 0.249i)12-s + 0.662·13-s − 1.00·14-s + 0.458·15-s + 0.250·16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(10.6839\)
Root analytic conductor: \(3.26863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1338} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.850589325\)
\(L(\frac12)\) \(\approx\) \(2.850589325\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 + 0.866i)T \)
223 \( 1 + (-0.344 - 14.9i)T \)
good5 \( 1 + (-0.887 - 1.53i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.74T + 7T^{2} \)
11 \( 1 + (-2.36 - 4.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 + (-1.21 + 2.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.36 - 7.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.90 + 5.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.83 - 8.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.632T + 41T^{2} \)
43 \( 1 + (-3.14 + 5.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.30 - 9.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.401 + 0.696i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.772T + 59T^{2} \)
61 \( 1 + (2.78 - 4.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.20 - 3.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.965 + 1.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.22 + 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.71 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.19 + 14.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.04 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.85 + 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.851890550760136608289602606620, −9.036608446154786717972951469708, −7.65167802065565968503732022888, −7.12598777300843490189503784739, −6.22908447545736102799721504710, −5.90295239529174388685262990060, −4.38935072957198022018806350038, −3.39620854930495144077652322501, −2.75148973259559424784817337585, −1.47106677811560570660229232664, 1.00790062268989365370244675447, 2.73256561401960417325053801614, 3.55531829626207913583425895240, 4.20968913239822422599566713431, 5.57582054343624408050957891503, 5.97757270996070163387887922125, 6.80127430246252888236425421854, 8.143925015849372814187698619998, 8.800502110827945271440335254162, 9.617415006959794384021021241693

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