Properties

Label 2-418-11.4-c1-0-13
Degree $2$
Conductor $418$
Sign $0.997 + 0.0692i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.0373 − 0.115i)3-s + (0.309 + 0.951i)4-s + (1.75 − 1.27i)5-s + (0.0978 − 0.0711i)6-s + (−1.23 − 3.79i)7-s + (−0.309 + 0.951i)8-s + (2.41 + 1.75i)9-s + 2.16·10-s + (3.30 + 0.308i)11-s + 0.120·12-s + (−4.23 − 3.07i)13-s + (1.23 − 3.79i)14-s + (−0.0810 − 0.249i)15-s + (−0.809 + 0.587i)16-s + (0.125 − 0.0911i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.0215 − 0.0664i)3-s + (0.154 + 0.475i)4-s + (0.784 − 0.570i)5-s + (0.0399 − 0.0290i)6-s + (−0.465 − 1.43i)7-s + (−0.109 + 0.336i)8-s + (0.805 + 0.584i)9-s + 0.685·10-s + (0.995 + 0.0929i)11-s + 0.0349·12-s + (−1.17 − 0.853i)13-s + (0.329 − 1.01i)14-s + (−0.0209 − 0.0644i)15-s + (−0.202 + 0.146i)16-s + (0.0304 − 0.0221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.997 + 0.0692i$
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: \((2,\ 418,\ (\ :1/2),\ 0.997 + 0.0692i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12759 - 0.0737681i\)
\(L(\frac12)\) \(\approx\) \(2.12759 - 0.0737681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 0.308i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.0373 + 0.115i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.75 + 1.27i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.23 + 3.79i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.23 + 3.07i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.125 + 0.0911i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 + (-1.66 - 5.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.64 + 1.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.17 - 6.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.28 - 10.1i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.72T + 43T^{2} \)
47 \( 1 + (-2.05 + 6.32i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.73 + 3.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.34 + 4.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.21 + 1.60i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.66T + 67T^{2} \)
71 \( 1 + (2.97 - 2.16i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.06 + 3.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.416 - 0.302i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.2 - 8.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 + (9.81 + 7.12i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.19530400073357544661441049707, −10.06649699985523724182863116632, −9.654145325804138703184913830122, −8.247791629738322051145692450714, −7.16906153328475438299120022868, −6.68745546959956763985926472135, −5.23061951344185429203433393660, −4.53155836175978671362406679447, −3.27332816963493076962960294088, −1.42034559725598926367097624289, 1.92842965778385797782818787098, 2.91497155044441044739134445790, 4.26102898875002286166370510471, 5.48401741667663329844558766926, 6.44101507540255065976060474447, 7.03053566347314426500967981644, 9.016486970454131668060656861364, 9.422834496574666659246809338413, 10.21611700150881480192287722587, 11.44187101446984935249109933616

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