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Properties

Label 2-416-13.12-c1-0-12
Degree $2$
Conductor $416$
Sign $-0.832 - 0.554i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3i·5-s + i·7-s + 6·9-s − 4i·11-s + (−2 + 3i)13-s + 9i·15-s − 5·17-s + 6i·19-s − 3i·21-s − 6·23-s − 4·25-s − 9·27-s − 4·29-s + 12i·33-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34i·5-s + 0.377i·7-s + 2·9-s − 1.20i·11-s + (−0.554 + 0.832i)13-s + 2.32i·15-s − 1.21·17-s + 1.37i·19-s − 0.654i·21-s − 1.25·23-s − 0.800·25-s − 1.73·27-s − 0.742·29-s + 2.08i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (2 - 3i)T \)
good3 \( 1 + 3T + 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 3T + 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2iT - 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 11iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.90058415489492309293663788560, −9.867705803055531930194665540861, −8.924066083756116351624652023821, −7.944595749674271891760273873132, −6.47010685473241215068343474096, −5.83759866425758622327451266892, −4.96700719456017829763534961263, −4.11737137647606119361774060872, −1.58958060048972467946282313599, 0, 2.31848070166838050919277286340, 4.12187527325456957596345917235, 5.10019492592589482081221636138, 6.20214531995203480907610222723, 7.00599394138295271063141268226, 7.46646046416840431551234300796, 9.428483445700316367073144700927, 10.45065652592070170228992426050, 10.72301841796029202465661168657

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