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Properties

Label 2-416-13.8-c2-0-3
Degree $2$
Conductor $416$
Sign $-0.0964 - 0.995i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7 − 7i)5-s − 9·9-s + (12 + 5i)13-s + 30i·17-s + 73i·25-s + 40·29-s + (−23 + 23i)37-s + (−49 − 49i)41-s + (63 + 63i)45-s + 49i·49-s − 90·53-s + 22·61-s + (−49 − 119i)65-s + (−103 + 103i)73-s + 81·81-s + ⋯
L(s)  = 1  + (−1.40 − 1.40i)5-s − 9-s + (0.923 + 0.384i)13-s + 1.76i·17-s + 2.91i·25-s + 1.37·29-s + (−0.621 + 0.621i)37-s + (−1.19 − 1.19i)41-s + (1.40 + 1.40i)45-s + 0.999i·49-s − 1.69·53-s + 0.360·61-s + (−0.753 − 1.83i)65-s + (−1.41 + 1.41i)73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0964 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0964 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.0964 - 0.995i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ -0.0964 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.337453 + 0.371736i\)
\(L(\frac12)\) \(\approx\) \(0.337453 + 0.371736i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-12 - 5i)T \)
good3 \( 1 + 9T^{2} \)
5 \( 1 + (7 + 7i)T + 25iT^{2} \)
7 \( 1 - 49iT^{2} \)
11 \( 1 - 121iT^{2} \)
17 \( 1 - 30iT - 289T^{2} \)
19 \( 1 + 361iT^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 40T + 841T^{2} \)
31 \( 1 + 961iT^{2} \)
37 \( 1 + (23 - 23i)T - 1.36e3iT^{2} \)
41 \( 1 + (49 + 49i)T + 1.68e3iT^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 2.20e3iT^{2} \)
53 \( 1 + 90T + 2.80e3T^{2} \)
59 \( 1 - 3.48e3iT^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3iT^{2} \)
71 \( 1 + 5.04e3iT^{2} \)
73 \( 1 + (103 - 103i)T - 5.32e3iT^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 + 6.88e3iT^{2} \)
89 \( 1 + (41 - 41i)T - 7.92e3iT^{2} \)
97 \( 1 + (-137 - 137i)T + 9.40e3iT^{2} \)
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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

−11.41012445304543214122650028070, −10.52838748846064366779925804968, −8.998518068969086855493034632019, −8.469057703608157818500561116102, −7.955303076044463993183207242137, −6.48223321822479379400090335885, −5.35306127225151793049558822652, −4.28761243844372223322587229404, −3.43881269410762970536797088313, −1.31873714462970097180279669404, 0.23461819360420644172229273626, 2.84690447047710187454759625908, 3.40333079498714128439871200488, 4.79239182285770299085814992403, 6.22543251224299680260777824096, 7.04528389378919521813763069367, 7.956044444775796745770994872324, 8.697190444327680956824538648298, 10.08492185490885724146589830752, 10.98801662990072814601146941475

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