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Properties

Label 2-418-11.4-c1-0-7
Degree $2$
Conductor $418$
Sign $-0.109 - 0.993i$
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 + (−0.396 + 1.22i)3-s + (0.309 + 0.951i)4-s + (1.09 − 0.799i)5-s + (−1.03 + 0.754i)6-s + (0.979 + 3.01i)7-s + (−0.309 + 0.951i)8-s + (1.09 + 0.794i)9-s + 1.35·10-s + (−0.286 − 3.30i)11-s − 1.28·12-s + (−0.734 − 0.533i)13-s + (−0.979 + 3.01i)14-s + (0.539 + 1.65i)15-s + (−0.809 + 0.587i)16-s + (−2.29 + 1.66i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.229 + 0.705i)3-s + (0.154 + 0.475i)4-s + (0.491 − 0.357i)5-s + (−0.424 + 0.308i)6-s + (0.370 + 1.13i)7-s + (−0.109 + 0.336i)8-s + (0.364 + 0.264i)9-s + 0.429·10-s + (−0.0863 − 0.996i)11-s − 0.370·12-s + (−0.203 − 0.147i)13-s + (−0.261 + 0.805i)14-s + (0.139 + 0.428i)15-s + (−0.202 + 0.146i)16-s + (−0.556 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.109 - 0.993i$
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: \((2,\ 418,\ (\ :1/2),\ -0.109 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29190 + 1.44275i\)
\(L(\frac12)\) \(\approx\) \(1.29190 + 1.44275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.286 + 3.30i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.396 - 1.22i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.09 + 0.799i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.979 - 3.01i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.734 + 0.533i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.29 - 1.66i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + (1.53 + 4.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.34 - 4.60i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.93 + 5.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.33 + 10.2i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.172T + 43T^{2} \)
47 \( 1 + (1.78 - 5.48i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.9 + 7.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.248 - 0.765i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.09 + 5.88i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (-10.6 + 7.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.09 + 3.37i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.58 + 3.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.58 + 1.88i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 + (3.27 + 2.37i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.39963871612799372688490201496, −10.70347539268771819292339501412, −9.509509139579234930793000775268, −8.750004905807792204651042265242, −7.83792437010697106597707979427, −6.40903214757587815523999559872, −5.47691310254806696547500342193, −4.94774578146311222533026123721, −3.65496815426679802872303010740, −2.16188007768039675136075842647, 1.22407830019876411678379490936, 2.49581010963308470965622130496, 4.11542436103591130074043960107, 4.94147399960320750186973654018, 6.49180571126327360292384361837, 6.91902420452125110446908743623, 7.895328164032726753691692058487, 9.549347529489839139185724612134, 10.14967809678910074403521644473, 11.09525991020092172814009266184

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