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Properties

Label 2-416-32.21-c1-0-4
Degree $2$
Conductor $416$
Sign $-0.929 + 0.368i$
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  + (−0.493 + 1.32i)2-s + (1.01 − 0.419i)3-s + (−1.51 − 1.30i)4-s + (−1.26 + 3.06i)5-s + (0.0562 + 1.54i)6-s + (−0.757 − 0.757i)7-s + (2.48 − 1.35i)8-s + (−1.27 + 1.27i)9-s + (−3.43 − 3.19i)10-s + (−5.35 − 2.21i)11-s + (−2.08 − 0.690i)12-s + (0.382 + 0.923i)13-s + (1.37 − 0.629i)14-s + 3.63i·15-s + (0.578 + 3.95i)16-s + 4.42i·17-s + ⋯
L(s)  = 1  + (−0.348 + 0.937i)2-s + (0.584 − 0.242i)3-s + (−0.756 − 0.653i)4-s + (−0.566 + 1.36i)5-s + (0.0229 + 0.632i)6-s + (−0.286 − 0.286i)7-s + (0.876 − 0.480i)8-s + (−0.423 + 0.423i)9-s + (−1.08 − 1.00i)10-s + (−1.61 − 0.668i)11-s + (−0.600 − 0.199i)12-s + (0.106 + 0.256i)13-s + (0.367 − 0.168i)14-s + 0.937i·15-s + (0.144 + 0.989i)16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.929 + 0.368i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0879250 - 0.459892i\)
\(L(\frac12)\) \(\approx\) \(0.0879250 - 0.459892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.493 - 1.32i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
good3 \( 1 + (-1.01 + 0.419i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.26 - 3.06i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.757 + 0.757i)T + 7iT^{2} \)
11 \( 1 + (5.35 + 2.21i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 + (1.84 + 4.46i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.30 - 2.30i)T - 23iT^{2} \)
29 \( 1 + (-4.83 + 2.00i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.03T + 31T^{2} \)
37 \( 1 + (1.25 - 3.03i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.20 - 7.20i)T - 41iT^{2} \)
43 \( 1 + (2.37 + 0.981i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.59iT - 47T^{2} \)
53 \( 1 + (-9.03 - 3.74i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.81 + 4.37i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.32 + 1.79i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-6.15 + 2.54i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.53 - 5.53i)T + 71iT^{2} \)
73 \( 1 + (8.59 - 8.59i)T - 73iT^{2} \)
79 \( 1 - 7.19iT - 79T^{2} \)
83 \( 1 + (-3.96 - 9.56i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.57 - 2.57i)T + 89iT^{2} \)
97 \( 1 + 4.78T + 97T^{2} \)
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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

−11.32839242029512589671728080924, −10.65115824466677835446296839801, −9.983271835239808889002705531672, −8.457197005507111537124567492106, −8.095477695592259499108866134250, −7.16622256602311584914925553112, −6.38306658998642634088034468107, −5.21676614227267917731834353907, −3.67779506186302128395467432611, −2.54459557777677910376080298925, 0.29847046513722889623685218790, 2.28488471449049277310502117920, 3.45510735130441537155695621354, 4.59090305386043785297689449930, 5.45179514737022286917168761577, 7.49976564861194469419294701260, 8.395470082705262008465803432532, 8.791590954287980799698199829369, 9.814633187374381444352115199929, 10.49370001985986048820024276384

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