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Properties

Label 2-416-104.11-c1-0-7
Degree $2$
Conductor $416$
Sign $0.975 - 0.219i$
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  + (0.928 + 1.60i)3-s + (1.51 − 1.51i)5-s + (2.97 − 0.797i)7-s + (−0.223 + 0.386i)9-s + (−0.865 − 0.231i)11-s + (−0.159 − 3.60i)13-s + (3.84 + 1.03i)15-s + (1.05 + 0.610i)17-s + (−6.68 + 1.79i)19-s + (4.04 + 4.04i)21-s + (−0.433 − 0.751i)23-s + 0.390i·25-s + 4.74·27-s + (3.26 − 1.88i)29-s + (−5.06 + 5.06i)31-s + ⋯
L(s)  = 1  + (0.535 + 0.928i)3-s + (0.678 − 0.678i)5-s + (1.12 − 0.301i)7-s + (−0.0743 + 0.128i)9-s + (−0.260 − 0.0699i)11-s + (−0.0442 − 0.999i)13-s + (0.994 + 0.266i)15-s + (0.256 + 0.147i)17-s + (−1.53 + 0.410i)19-s + (0.883 + 0.883i)21-s + (−0.0904 − 0.156i)23-s + 0.0780i·25-s + 0.912·27-s + (0.606 − 0.350i)29-s + (−0.909 + 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.975 - 0.219i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94492 + 0.216075i\)
\(L(\frac12)\) \(\approx\) \(1.94492 + 0.216075i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.159 + 3.60i)T \)
good3 \( 1 + (-0.928 - 1.60i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.51 + 1.51i)T - 5iT^{2} \)
7 \( 1 + (-2.97 + 0.797i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.865 + 0.231i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.05 - 0.610i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.68 - 1.79i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.433 + 0.751i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.26 + 1.88i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.06 - 5.06i)T - 31iT^{2} \)
37 \( 1 + (2.52 - 9.43i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.12 - 11.6i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.22 - 2.44i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 + 2.16iT - 53T^{2} \)
59 \( 1 + (0.0382 + 0.142i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.26 + 4.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.422 - 1.57i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.27 + 15.9i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \)
79 \( 1 - 6.37iT - 79T^{2} \)
83 \( 1 + (3.53 + 3.53i)T + 83iT^{2} \)
89 \( 1 + (-9.33 - 2.50i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.6 - 2.86i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.88412257036020915924711732649, −10.31416530393535063949346672193, −9.484184866149994461080074400154, −8.461911619318723675417298077120, −8.017481286586547638788382159051, −6.38715453477412285320767931102, −5.12530979972171193359674368991, −4.52805714218046460337257854617, −3.21209303734983575832333975621, −1.57735721613116801150217085813, 1.88049241371162850629691024157, 2.40752838336305990880120502049, 4.27031566540695760861872321502, 5.55411239033599254338840227950, 6.69851526660139371633523873470, 7.40622444225036150606662708188, 8.388265408137165573412509129714, 9.141676156057092273566901358835, 10.46085532704549707229054679690, 11.08787039837525676122349875191

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