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Properties

Label 2-714-17.13-c1-0-12
Degree $2$
Conductor $714$
Sign $0.879 - 0.476i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
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  + i·2-s + (0.707 + 0.707i)3-s − 4-s + (−2.30 − 2.30i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)7-s − i·8-s + 1.00i·9-s + (2.30 − 2.30i)10-s + (1.30 − 1.30i)11-s + (−0.707 − 0.707i)12-s + 5.71·13-s + (−0.707 − 0.707i)14-s − 3.26i·15-s + 16-s + (1.64 − 3.77i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−1.03 − 1.03i)5-s + (−0.288 + 0.288i)6-s + (−0.267 + 0.267i)7-s − 0.353i·8-s + 0.333i·9-s + (0.729 − 0.729i)10-s + (0.393 − 0.393i)11-s + (−0.204 − 0.204i)12-s + 1.58·13-s + (−0.188 − 0.188i)14-s − 0.841i·15-s + 0.250·16-s + (0.399 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.879 - 0.476i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.879 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39545 + 0.354017i\)
\(L(\frac12)\) \(\approx\) \(1.39545 + 0.354017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-1.64 + 3.77i)T \)
good5 \( 1 + (2.30 + 2.30i)T + 5iT^{2} \)
11 \( 1 + (-1.30 + 1.30i)T - 11iT^{2} \)
13 \( 1 - 5.71T + 13T^{2} \)
19 \( 1 - 5.00iT - 19T^{2} \)
23 \( 1 + (-4.58 + 4.58i)T - 23iT^{2} \)
29 \( 1 + (-6.80 - 6.80i)T + 29iT^{2} \)
31 \( 1 + (0.764 + 0.764i)T + 31iT^{2} \)
37 \( 1 + (1.17 + 1.17i)T + 37iT^{2} \)
41 \( 1 + (-8.19 + 8.19i)T - 41iT^{2} \)
43 \( 1 + 6.49iT - 43T^{2} \)
47 \( 1 + 4.87T + 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 + (-5.77 + 5.77i)T - 61iT^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (-1.96 - 1.96i)T + 71iT^{2} \)
73 \( 1 + (-6.98 - 6.98i)T + 73iT^{2} \)
79 \( 1 + (5.84 - 5.84i)T - 79iT^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + (-3.89 - 3.89i)T + 97iT^{2} \)
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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

−10.40949581223917471616950074011, −9.141966346357453348438818794947, −8.708711068641555629964737499391, −8.149606297856298134715887806548, −7.10422162949242333646128884587, −5.97935184053618849402935264151, −5.01492783483610176904902161426, −4.05961335181483651888291302465, −3.28718849543678347207527244783, −0.957165640463134487487687134222, 1.18426477196190134429437411213, 2.84550404615348220698302826238, 3.55443718377146751285287298887, 4.39458439364771011187987417555, 6.14719908098918774565003487232, 6.91415312031766194305481699400, 7.84980265234682434115289828332, 8.578808028593953949272777617675, 9.591551514676736874702159271519, 10.53702692409146537263640269882

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