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Properties

Label 2-416-416.389-c1-0-19
Degree $2$
Conductor $416$
Sign $0.982 - 0.186i$
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  + (−0.988 − 1.01i)2-s + (0.708 + 1.71i)3-s + (−0.0474 + 1.99i)4-s + (0.188 − 0.454i)5-s + (1.03 − 2.40i)6-s + (0.461 − 0.461i)7-s + (2.06 − 1.92i)8-s + (−0.301 + 0.301i)9-s + (−0.645 + 0.258i)10-s + (1.38 + 0.572i)11-s + (−3.45 + 1.33i)12-s + (2.41 − 2.67i)13-s + (−0.923 − 0.0109i)14-s + 0.910·15-s + (−3.99 − 0.189i)16-s + 1.70i·17-s + ⋯
L(s)  = 1  + (−0.698 − 0.715i)2-s + (0.408 + 0.987i)3-s + (−0.0237 + 0.999i)4-s + (0.0842 − 0.203i)5-s + (0.420 − 0.982i)6-s + (0.174 − 0.174i)7-s + (0.731 − 0.681i)8-s + (−0.100 + 0.100i)9-s + (−0.204 + 0.0817i)10-s + (0.416 + 0.172i)11-s + (−0.996 + 0.385i)12-s + (0.670 − 0.741i)13-s + (−0.246 − 0.00292i)14-s + 0.235·15-s + (−0.998 − 0.0474i)16-s + 0.413i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.982 - 0.186i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22214 + 0.114886i\)
\(L(\frac12)\) \(\approx\) \(1.22214 + 0.114886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.988 + 1.01i)T \)
13 \( 1 + (-2.41 + 2.67i)T \)
good3 \( 1 + (-0.708 - 1.71i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.188 + 0.454i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.461 + 0.461i)T - 7iT^{2} \)
11 \( 1 + (-1.38 - 0.572i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 - 1.70iT - 17T^{2} \)
19 \( 1 + (-1.71 - 4.14i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.296 - 0.296i)T - 23iT^{2} \)
29 \( 1 + (0.673 + 1.62i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 5.90iT - 31T^{2} \)
37 \( 1 + (1.12 - 2.70i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.52 + 6.52i)T + 41iT^{2} \)
43 \( 1 + (0.359 - 0.867i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.98T + 47T^{2} \)
53 \( 1 + (-4.36 + 10.5i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.83 + 9.25i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.18 + 7.69i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.91 - 1.61i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-3.34 + 3.34i)T - 71iT^{2} \)
73 \( 1 + (7.51 + 7.51i)T + 73iT^{2} \)
79 \( 1 + 2.03iT - 79T^{2} \)
83 \( 1 + (-0.942 - 2.27i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.61 - 8.61i)T - 89iT^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−10.86186295799140824890276793147, −10.31564953504057592368739429077, −9.521851260919626720520068105664, −8.735241869702819604437237956362, −7.993387877146409583040427118531, −6.74534173821553678015793080185, −5.14902960430581636287445597482, −3.91232153343888182658347863929, −3.25760406126560837113114167055, −1.44188101091527142379886141606, 1.21323183950682249797627040483, 2.50758024008050730119770070165, 4.49506712798101561178187196677, 5.85355052047600230891143403635, 6.82958385175513779912473916956, 7.33459095602722421818548040953, 8.470212735133976992246101536012, 8.987658755549027064281887443683, 10.09929704853876231230530183202, 11.15127337385058209273504429190

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