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Properties

Label 2-416-104.99-c1-0-1
Degree $2$
Conductor $416$
Sign $-0.267 - 0.963i$
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  − 2.19·3-s + (0.274 − 0.274i)5-s + (0.352 + 0.352i)7-s + 1.81·9-s + (0.275 − 0.275i)11-s + (−3.16 + 1.72i)13-s + (−0.602 + 0.602i)15-s + 2.40i·17-s + (4.12 + 4.12i)19-s + (−0.773 − 0.773i)21-s − 6.72·23-s + 4.84i·25-s + 2.60·27-s + 8.97i·29-s + (−5.14 + 5.14i)31-s + ⋯
L(s)  = 1  − 1.26·3-s + (0.122 − 0.122i)5-s + (0.133 + 0.133i)7-s + 0.604·9-s + (0.0831 − 0.0831i)11-s + (−0.877 + 0.479i)13-s + (−0.155 + 0.155i)15-s + 0.582i·17-s + (0.946 + 0.946i)19-s + (−0.168 − 0.168i)21-s − 1.40·23-s + 0.969i·25-s + 0.501·27-s + 1.66i·29-s + (−0.923 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342688 + 0.450829i\)
\(L(\frac12)\) \(\approx\) \(0.342688 + 0.450829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 - 1.72i)T \)
good3 \( 1 + 2.19T + 3T^{2} \)
5 \( 1 + (-0.274 + 0.274i)T - 5iT^{2} \)
7 \( 1 + (-0.352 - 0.352i)T + 7iT^{2} \)
11 \( 1 + (-0.275 + 0.275i)T - 11iT^{2} \)
17 \( 1 - 2.40iT - 17T^{2} \)
19 \( 1 + (-4.12 - 4.12i)T + 19iT^{2} \)
23 \( 1 + 6.72T + 23T^{2} \)
29 \( 1 - 8.97iT - 29T^{2} \)
31 \( 1 + (5.14 - 5.14i)T - 31iT^{2} \)
37 \( 1 + (3.81 + 3.81i)T + 37iT^{2} \)
41 \( 1 + (-1.19 - 1.19i)T + 41iT^{2} \)
43 \( 1 - 5.40iT - 43T^{2} \)
47 \( 1 + (-2.83 - 2.83i)T + 47iT^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + (0.881 - 0.881i)T - 59iT^{2} \)
61 \( 1 - 2.65iT - 61T^{2} \)
67 \( 1 + (-1.69 - 1.69i)T + 67iT^{2} \)
71 \( 1 + (-9.28 + 9.28i)T - 71iT^{2} \)
73 \( 1 + (1.19 - 1.19i)T - 73iT^{2} \)
79 \( 1 - 8.08iT - 79T^{2} \)
83 \( 1 + (8.49 + 8.49i)T + 83iT^{2} \)
89 \( 1 + (-10.2 + 10.2i)T - 89iT^{2} \)
97 \( 1 + (-6.14 - 6.14i)T + 97iT^{2} \)
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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

−11.55039405883972805371697490925, −10.66751662877699964841325789507, −9.876800951383570600967813766173, −8.837731314396092250665512275322, −7.60941941201450925623255739218, −6.65403950918608777023278420944, −5.61116077586939935857507953762, −5.03057256403443202380235200762, −3.60279034245119838734972068942, −1.65287169304291665074735012794, 0.42563834523289868350937637631, 2.51267259188176278907973266816, 4.26895573726387778020907997908, 5.27656458284055788999626455182, 6.04747327979744636124331427993, 7.09653176924078593375162146477, 7.976720875643199010550589495466, 9.429024051508372228114137436319, 10.15856731658368772762196047204, 11.04073486588699558671104578055

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