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Properties

Label 2-416-52.35-c2-0-24
Degree $2$
Conductor $416$
Sign $-0.0566 + 0.998i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.02 + 1.17i)3-s − 2.43·5-s + (−0.984 + 0.568i)7-s + (−1.75 − 3.04i)9-s + (−16.8 − 9.75i)11-s + (7.05 − 10.9i)13-s + (−4.94 − 2.85i)15-s + (−11.0 − 19.1i)17-s + (26.0 − 15.0i)19-s − 2.66·21-s + (32.0 + 18.4i)23-s − 19.0·25-s − 29.3i·27-s + (−21.5 + 37.3i)29-s − 15.3i·31-s + ⋯
L(s)  = 1  + (0.676 + 0.390i)3-s − 0.487·5-s + (−0.140 + 0.0812i)7-s + (−0.195 − 0.337i)9-s + (−1.53 − 0.886i)11-s + (0.542 − 0.839i)13-s + (−0.329 − 0.190i)15-s + (−0.649 − 1.12i)17-s + (1.37 − 0.791i)19-s − 0.126·21-s + (1.39 + 0.804i)23-s − 0.762·25-s − 1.08i·27-s + (−0.744 + 1.28i)29-s − 0.493i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0566 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0566 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.0566 + 0.998i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ -0.0566 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.847929 - 0.897400i\)
\(L(\frac12)\) \(\approx\) \(0.847929 - 0.897400i\)
\(L(2)\) 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 \( 1 + (-7.05 + 10.9i)T \)
good3 \( 1 + (-2.02 - 1.17i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + 2.43T + 25T^{2} \)
7 \( 1 + (0.984 - 0.568i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (16.8 + 9.75i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + (11.0 + 19.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-26.0 + 15.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-32.0 - 18.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (21.5 - 37.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 15.3iT - 961T^{2} \)
37 \( 1 + (-25.7 + 44.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (5.36 - 9.29i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-8.04 + 4.64i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 27.4iT - 2.20e3T^{2} \)
53 \( 1 + 73.3T + 2.80e3T^{2} \)
59 \( 1 + (-18.2 + 10.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.0 + 91.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (4.45 + 2.57i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-13.0 + 7.51i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 3.70T + 5.32e3T^{2} \)
79 \( 1 - 78.9iT - 6.24e3T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 + (-17.6 + 30.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.2 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.08447437446303142651552313303, −9.586654516018336928247168109917, −9.060194189022136605541437499263, −7.989653103658062547897543927374, −7.34520026871542455360932238760, −5.80539943570397228528674369243, −4.93469910238129144689383927406, −3.31534479187639542873104283845, −2.92469092668267896066946056366, −0.47193744169485653733999793137, 1.81556364251547444770341907619, 2.97591497749907345135853011887, 4.26665009097691598474286558558, 5.42102737651659975988605370125, 6.76788140613822291445398303414, 7.78039422620599983581401365341, 8.205192267266821015297496550036, 9.340052872456811022269046256462, 10.35420265391857544695916766564, 11.19960521854214193078174880970

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