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Properties

Label 2-416-416.77-c1-0-28
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: $\chi_{416} (77, \cdot )$
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

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

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