↑

Properties

Label 2-416-416.83-c1-0-4
Degree $2$
Conductor $416$
Sign $0.648 - 0.760i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 0.0832i)2-s + (−2.32 − 0.961i)3-s + (1.98 + 0.235i)4-s + (0.932 − 2.25i)5-s + (3.19 + 1.55i)6-s + 4.08i·7-s + (−2.78 − 0.497i)8-s + (2.34 + 2.34i)9-s + (−1.50 + 3.10i)10-s + (−5.01 − 2.07i)11-s + (−4.38 − 2.45i)12-s + (3.14 + 1.75i)13-s + (0.339 − 5.76i)14-s + (−4.33 + 4.33i)15-s + (3.88 + 0.934i)16-s − 5.02·17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0588i)2-s + (−1.34 − 0.555i)3-s + (0.993 + 0.117i)4-s + (0.417 − 1.00i)5-s + (1.30 + 0.633i)6-s + 1.54i·7-s + (−0.984 − 0.175i)8-s + (0.781 + 0.781i)9-s + (−0.475 + 0.980i)10-s + (−1.51 − 0.626i)11-s + (−1.26 − 0.709i)12-s + (0.873 + 0.487i)13-s + (0.0908 − 1.53i)14-s + (−1.11 + 1.11i)15-s + (0.972 + 0.233i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.648 - 0.760i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.648 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367895 + 0.169752i\)
\(L(\frac12)\) \(\approx\) \(0.367895 + 0.169752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0832i)T \)
13 \( 1 + (-3.14 - 1.75i)T \)
good3 \( 1 + (2.32 + 0.961i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.932 + 2.25i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 - 4.08iT - 7T^{2} \)
11 \( 1 + (5.01 + 2.07i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 + 5.02T + 17T^{2} \)
19 \( 1 + (0.314 + 0.759i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.64 - 5.64i)T + 23iT^{2} \)
29 \( 1 + (1.61 - 3.89i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.191 - 0.191i)T + 31iT^{2} \)
37 \( 1 + (-4.82 - 1.99i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
43 \( 1 + (0.182 + 0.440i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (-6.13 - 6.13i)T + 47iT^{2} \)
53 \( 1 + (-3.34 - 8.06i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.901 - 2.17i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.967 + 2.33i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.21 + 0.503i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + 1.22T + 71T^{2} \)
73 \( 1 + 4.96iT - 73T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + (1.66 + 4.01i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 5.69T + 89T^{2} \)
97 \( 1 + (6.00 - 6.00i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.19358897592591591987917242355, −10.77247501969833758091282689248, −9.092777153724654105984658612354, −8.961156980507726396799670378203, −7.75322008854259537956945968303, −6.48552843102054287656884714284, −5.69966280866006985640495185541, −5.15707925273424879032668574601, −2.60676737478445037974093800382, −1.23717439021483633095278275796, 0.46567380299628967764169754387, 2.59855126748119942754930928688, 4.27847190524267146618305842473, 5.56999320009487803520885237169, 6.59991580746071293194865023816, 7.13866454408819402721525954795, 8.253710210125213498670229491518, 9.796309165881864644850422045952, 10.45500037885933247296167498573, 10.82623568582672809717776569655

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