Properties

Label 2-418-11.4-c1-0-17
Degree $2$
Conductor $418$
Sign $0.231 + 0.972i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (1.02 − 3.14i)3-s + (0.309 + 0.951i)4-s + (0.670 − 0.487i)5-s + (2.67 − 1.94i)6-s + (−0.688 − 2.11i)7-s + (−0.309 + 0.951i)8-s + (−6.42 − 4.66i)9-s + 0.828·10-s + (0.689 + 3.24i)11-s + 3.30·12-s + (0.976 + 0.709i)13-s + (0.688 − 2.11i)14-s + (−0.847 − 2.60i)15-s + (−0.809 + 0.587i)16-s + (3.70 − 2.69i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.590 − 1.81i)3-s + (0.154 + 0.475i)4-s + (0.299 − 0.217i)5-s + (1.09 − 0.793i)6-s + (−0.260 − 0.801i)7-s + (−0.109 + 0.336i)8-s + (−2.14 − 1.55i)9-s + 0.262·10-s + (0.207 + 0.978i)11-s + 0.954·12-s + (0.270 + 0.196i)13-s + (0.184 − 0.566i)14-s + (−0.218 − 0.673i)15-s + (−0.202 + 0.146i)16-s + (0.899 − 0.653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73984 - 1.37506i\)
\(L(\frac12)\) \(\approx\) \(1.73984 - 1.37506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.689 - 3.24i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-1.02 + 3.14i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.670 + 0.487i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.688 + 2.11i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.976 - 0.709i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.70 + 2.69i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + 1.76T + 23T^{2} \)
29 \( 1 + (1.20 + 3.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.02 - 5.10i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.423 + 1.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.82 + 5.62i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.78T + 43T^{2} \)
47 \( 1 + (2.44 - 7.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.58 - 4.78i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0323 + 0.0994i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-10.1 + 7.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + (4.36 - 3.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.09 - 12.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.87 + 1.36i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.84 - 3.51i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.07T + 89T^{2} \)
97 \( 1 + (14.0 + 10.1i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.50080896568557776058377450456, −9.945867868579326148677957582401, −8.901185325284198235621940219233, −7.81585220369433311060510571432, −7.26313942023310575963413345002, −6.54023206528354212273800630628, −5.53644926527062409125183623375, −3.92636224230624288318406605364, −2.62230777685407939139633175225, −1.28803283484555264954634437764, 2.57371333283428744307128441767, 3.38936789129391791949998502194, 4.31647212587368318153838993420, 5.53343431960702723780379330775, 6.07499712676102310725425058667, 8.198300680599022066314964162368, 8.867703773890097271624034296773, 9.851676059475983835676278971078, 10.34611436723617489696467109271, 11.24705916459290239721021146581

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