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Properties

Label 2-416-13.10-c1-0-9
Degree $2$
Conductor $416$
Sign $0.998 - 0.0612i$
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  + (1.13 + 1.96i)3-s − 3.99i·5-s + (0.981 + 0.566i)7-s + (−1.06 + 1.84i)9-s + (0.981 − 0.566i)11-s + (3.59 − 0.266i)13-s + (7.84 − 4.52i)15-s + (−0.5 + 0.866i)17-s + (3.39 + 1.96i)19-s + 2.56i·21-s + (−4.59 − 7.96i)23-s − 10.9·25-s + 1.96·27-s + (2.02 + 3.51i)29-s + 9.05i·31-s + ⋯
L(s)  = 1  + (0.654 + 1.13i)3-s − 1.78i·5-s + (0.371 + 0.214i)7-s + (−0.355 + 0.615i)9-s + (0.295 − 0.170i)11-s + (0.997 − 0.0740i)13-s + (2.02 − 1.16i)15-s + (−0.121 + 0.210i)17-s + (0.779 + 0.450i)19-s + 0.560i·21-s + (−0.958 − 1.66i)23-s − 2.19·25-s + 0.377·27-s + (0.376 + 0.652i)29-s + 1.62i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.998 - 0.0612i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.998 - 0.0612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81834 + 0.0557048i\)
\(L(\frac12)\) \(\approx\) \(1.81834 + 0.0557048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.266i)T \)
good3 \( 1 + (-1.13 - 1.96i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.99iT - 5T^{2} \)
7 \( 1 + (-0.981 - 0.566i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.981 + 0.566i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.39 - 1.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.59 + 7.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.02 - 3.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.05iT - 31T^{2} \)
37 \( 1 + (-2.16 + 1.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.42 - 1.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.79iT - 47T^{2} \)
53 \( 1 + 8.92T + 53T^{2} \)
59 \( 1 + (7.77 + 4.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.89 + 6.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.36 + 4.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.86 + 3.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.29iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 5.73iT - 83T^{2} \)
89 \( 1 + (-12.1 + 7.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.70 + 4.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.12640565443079082999039100949, −10.06877135982827382742497263585, −9.290758525056595540190192852567, −8.457683577673056491548455600549, −8.258394996047179089664195697110, −6.26384206570283884151823333641, −5.04164445023810159678677999622, −4.43189289584267023071313694442, −3.37982428025957593587301718800, −1.40368495723740351954214261366, 1.72749934854714382428168552435, 2.84628235140292911840868361916, 3.87942500606346993520523132255, 5.88006213293885188171955398702, 6.73985050305483242111909637722, 7.49357857963855507128056892914, 8.041108884766681489306667253579, 9.409444524856461182600450219914, 10.36476672115106593816321706736, 11.41312871654361782642305229882

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