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Properties

Label 2-1338-223.183-c1-0-10
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 + (−1.69 + 2.92i)11-s + (0.5 + 0.866i)12-s + 5.47·13-s − 2.23·14-s − 0.999·15-s + 16-s − 1.14·17-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 + (−0.509 + 0.883i)11-s + (0.144 + 0.249i)12-s + 1.51·13-s − 0.597·14-s − 0.258·15-s + 0.250·16-s − 0.277·17-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: $\chi_{1338} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1338,\ (\ :1/2),\ -0.282 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.224652388\)
\(L(\frac12)\) \(\approx\) \(2.224652388\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 2.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 + (-1.19 - 2.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.11 + 1.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.16 - 7.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.38 - 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.59 - 9.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.85T + 41T^{2} \)
43 \( 1 + (1.54 + 2.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.04 - 1.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 + (-2.38 - 4.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.309 - 0.535i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.04 + 3.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.736 - 1.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.54 + 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.19 - 7.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.78 + 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.18709651488908596688007316460, −9.007519091225434965254939547219, −8.313779234516428663038689954328, −7.14581017569050487615155681620, −6.61832487878401388514131451482, −5.57197384063155249128602662407, −4.73609100342162121471543736328, −3.53072746166489108217237558415, −3.26634853653577727539520073599, −1.80772605484671036595949622172, 0.68567420780539477397549194449, 2.24572059500548664970303545371, 3.36720697879973723830358758439, 3.95440559093252370063715521985, 5.30536312778315367400024053551, 6.09537003045511297673827332622, 6.69145304258922861698638320497, 7.78225763221678637116633293999, 8.468449116084071642718215811358, 9.223677021750097775092106973787

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