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Properties

Label 2-416-416.411-c1-0-18
Degree $2$
Conductor $416$
Sign $0.704 - 0.709i$
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  + (−1.24 − 0.672i)2-s + (1.34 − 0.555i)3-s + (1.09 + 1.67i)4-s + (1.54 + 3.73i)5-s + (−2.04 − 0.211i)6-s + 0.167i·7-s + (−0.234 − 2.81i)8-s + (−0.632 + 0.632i)9-s + (0.589 − 5.68i)10-s + (0.115 − 0.0478i)11-s + (2.39 + 1.63i)12-s + (0.804 + 3.51i)13-s + (0.112 − 0.208i)14-s + (4.14 + 4.14i)15-s + (−1.60 + 3.66i)16-s − 4.75·17-s + ⋯
L(s)  = 1  + (−0.879 − 0.475i)2-s + (0.773 − 0.320i)3-s + (0.547 + 0.837i)4-s + (0.691 + 1.67i)5-s + (−0.833 − 0.0863i)6-s + 0.0632i·7-s + (−0.0829 − 0.996i)8-s + (−0.210 + 0.210i)9-s + (0.186 − 1.79i)10-s + (0.0348 − 0.0144i)11-s + (0.691 + 0.472i)12-s + (0.223 + 0.974i)13-s + (0.0301 − 0.0556i)14-s + (1.07 + 1.07i)15-s + (−0.401 + 0.915i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13388 + 0.472203i\)
\(L(\frac12)\) \(\approx\) \(1.13388 + 0.472203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.672i)T \)
13 \( 1 + (-0.804 - 3.51i)T \)
good3 \( 1 + (-1.34 + 0.555i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.54 - 3.73i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 - 0.167iT - 7T^{2} \)
11 \( 1 + (-0.115 + 0.0478i)T + (7.77 - 7.77i)T^{2} \)
17 \( 1 + 4.75T + 17T^{2} \)
19 \( 1 + (-1.08 + 2.62i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.822 - 0.822i)T - 23iT^{2} \)
29 \( 1 + (0.362 + 0.876i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.60 + 1.60i)T - 31iT^{2} \)
37 \( 1 + (-9.94 + 4.11i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 + (-2.99 + 7.23i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (-5.14 + 5.14i)T - 47iT^{2} \)
53 \( 1 + (-0.952 + 2.29i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.80 - 11.6i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.39 + 3.37i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-5.27 - 2.18i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 8.93iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-4.24 + 10.2i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 6.33T + 89T^{2} \)
97 \( 1 + (7.29 + 7.29i)T + 97iT^{2} \)
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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

−11.08839064998793446157461530321, −10.49156991744301373802282762928, −9.395764278324130480670248586605, −8.845259520242848239469663506617, −7.58074702816235839169922247761, −6.97811199170468831583591738910, −6.06735388008582330289790731998, −3.87786080962000735280154133621, −2.59577277094705939570397256442, −2.15837838765190770317265431945, 0.989540250095804174640263152590, 2.50400782621931479362899056075, 4.35277405225942873493620876199, 5.49830973588569366581058052535, 6.29745707851370418007694637281, 7.954155468348914991594762960124, 8.409746869841275489189698094703, 9.274229733122500580553914362311, 9.665345619885795411471800111371, 10.76423921269075994541411564384

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