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Properties

Label 2-1338-223.39-c1-0-34
Degree $2$
Conductor $1338$
Sign $-0.282 + 0.959i$
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 + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + 2.23·7-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−2.80 − 4.86i)11-s + (0.5 − 0.866i)12-s − 3.47·13-s + 2.23·14-s − 0.999·15-s + 16-s − 7.85·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + 0.845·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.846 − 1.46i)11-s + (0.144 − 0.249i)12-s − 0.962·13-s + 0.597·14-s − 0.258·15-s + 0.250·16-s − 1.90·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1338\)    =    \(2 \cdot 3 \cdot 223\)
Sign: $-0.282 + 0.959i$
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: \((2,\ 1338,\ (\ :1/2),\ -0.282 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.423167223\)
\(L(\frac12)\) \(\approx\) \(2.423167223\)
\(L(\frac{3}{2})\) 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 \( 1 + (11.5 + 9.52i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.23T + 7T^{2} \)
11 \( 1 + (2.80 + 4.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.47T + 13T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 + (-2.30 + 3.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.66 - 6.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.61 + 6.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.59 - 9.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.14T + 41T^{2} \)
43 \( 1 + (-4.04 + 7.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.54 - 7.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 + (-4.61 + 7.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.809 - 1.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.54 + 6.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.73 - 6.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.954 - 1.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.85 + 3.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.30 + 9.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.28 + 3.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.085652186312195457693201396868, −8.448033858647660926241635059274, −7.78939262179930997966682212627, −6.87079672615030637982454649612, −6.05232707413682134743045865303, −4.88122762061427243653906547376, −4.55283970957355774966826563131, −2.98740397640824421570345438476, −2.35363794275409623249494708338, −0.72549131070420682599720093445, 2.02322793634105521370595954223, 2.70152366737079461801888677052, 4.06665257577235206172875898509, 4.72361342419672367260784959305, 5.30758398666087911651881001972, 6.64779817855691842453638241946, 7.44970294864970895501764698451, 7.987163892539744204903839900250, 9.147592467978219766140929102624, 10.00060689605799928981848455297

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