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Properties

Label 2-416-416.389-c1-0-28
Degree $2$
Conductor $416$
Sign $-0.507 + 0.861i$
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  + (−1.40 − 0.159i)2-s + (−0.700 − 1.69i)3-s + (1.94 + 0.446i)4-s + (−0.175 + 0.423i)5-s + (0.715 + 2.48i)6-s + (−0.506 + 0.506i)7-s + (−2.66 − 0.938i)8-s + (−0.248 + 0.248i)9-s + (0.314 − 0.567i)10-s + (2.80 + 1.16i)11-s + (−0.609 − 3.61i)12-s + (0.593 − 3.55i)13-s + (0.792 − 0.631i)14-s + 0.839·15-s + (3.60 + 1.74i)16-s − 7.47i·17-s + ⋯
L(s)  = 1  + (−0.993 − 0.112i)2-s + (−0.404 − 0.976i)3-s + (0.974 + 0.223i)4-s + (−0.0785 + 0.189i)5-s + (0.292 + 1.01i)6-s + (−0.191 + 0.191i)7-s + (−0.943 − 0.331i)8-s + (−0.0828 + 0.0828i)9-s + (0.0993 − 0.179i)10-s + (0.846 + 0.350i)11-s + (−0.176 − 1.04i)12-s + (0.164 − 0.986i)13-s + (0.211 − 0.168i)14-s + 0.216·15-s + (0.900 + 0.435i)16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.507 + 0.861i$
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: \((2,\ 416,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333128 - 0.583014i\)
\(L(\frac12)\) \(\approx\) \(0.333128 - 0.583014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.159i)T \)
13 \( 1 + (-0.593 + 3.55i)T \)
good3 \( 1 + (0.700 + 1.69i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.175 - 0.423i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.506 - 0.506i)T - 7iT^{2} \)
11 \( 1 + (-2.80 - 1.16i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 + 7.47iT - 17T^{2} \)
19 \( 1 + (0.221 + 0.534i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.70 - 4.70i)T - 23iT^{2} \)
29 \( 1 + (1.96 + 4.73i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 + (-3.18 + 7.67i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.59 + 5.59i)T + 41iT^{2} \)
43 \( 1 + (-1.85 + 4.47i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (1.65 - 3.98i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.82 + 11.6i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.04 + 2.52i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-12.9 + 5.37i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (3.76 - 3.76i)T - 71iT^{2} \)
73 \( 1 + (-1.03 - 1.03i)T + 73iT^{2} \)
79 \( 1 - 2.10iT - 79T^{2} \)
83 \( 1 + (-5.22 - 12.6i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.45 + 6.45i)T - 89iT^{2} \)
97 \( 1 + 2.02iT - 97T^{2} \)
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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

−11.08807464611049319227874710279, −9.761861507807961477018982797998, −9.287572387263650618227531207536, −7.927607503482622917481655861557, −7.27607549775093402600731171122, −6.52854337954516492769506036421, −5.52416965282612094602832494059, −3.51597650533044035758303495335, −2.09741189571224280114593523086, −0.64603746695543579940546056549, 1.63777839807669993434384088859, 3.60612185122130928875706942740, 4.63387768119197603846500649468, 6.11817441475412030030817614179, 6.70280822754735648083913213154, 8.209421667012141283991905330338, 8.800856223965595222102736777009, 9.885595707190786687911253890461, 10.37245817815333209496840148704, 11.24040562811393388771916713888

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