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Properties

Label 2-416-104.101-c1-0-9
Degree $2$
Conductor $416$
Sign $0.869 + 0.494i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (2.36 − 1.36i)3-s − 0.267·5-s + (3 + 1.73i)7-s + (2.23 − 3.86i)9-s + (1 + 1.73i)11-s + (−2.59 − 2.5i)13-s + (−0.633 + 0.366i)15-s + (−3.23 + 5.59i)17-s + (2.36 − 4.09i)19-s + 9.46·21-s + (−1.09 − 1.90i)23-s − 4.92·25-s − 4.00i·27-s + (2.59 − 1.5i)29-s + 1.26i·31-s + ⋯
L(s)  = 1  + (1.36 − 0.788i)3-s − 0.119·5-s + (1.13 + 0.654i)7-s + (0.744 − 1.28i)9-s + (0.301 + 0.522i)11-s + (−0.720 − 0.693i)13-s + (−0.163 + 0.0945i)15-s + (−0.783 + 1.35i)17-s + (0.542 − 0.940i)19-s + 2.06·21-s + (−0.228 − 0.396i)23-s − 0.985·25-s − 0.769i·27-s + (0.482 − 0.278i)29-s + 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14387 - 0.567088i\)
\(L(\frac12)\) \(\approx\) \(2.14387 - 0.567088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (2.59 + 2.5i)T \)
good3 \( 1 + (-2.36 + 1.36i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.267T + 5T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (3.86 + 6.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.03 - 0.598i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.19 + 4.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.26iT - 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 + (-3.73 + 6.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 - 6.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 5.46T + 83T^{2} \)
89 \( 1 + (0.464 - 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27762376141748608947925103301, −10.06893405464887217921104285338, −8.970989151746805224911745052336, −8.346373009948488024902477988936, −7.67534564828097902615337365244, −6.74743233091417246276279002591, −5.30055637005559759203860821303, −4.04392846819313277584010069805, −2.58420996050725076309309086862, −1.76629636397938933427033842376, 1.89807204594735360958070209626, 3.28787758266785757970085005025, 4.27949759026825404671493706634, 5.08458303967688827187535036294, 6.91078046821121709237118430380, 7.88657552138808743317443178834, 8.494390710908247392814130186058, 9.528093481241718164448345801617, 10.07419271348816401205714715288, 11.32003498994018216089960939791

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