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Properties

Label 2-714-17.16-c1-0-11
Degree $2$
Conductor $714$
Sign $0.575 + 0.817i$
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  − 2-s + i·3-s + 4-s − 3.51i·5-s − i·6-s − i·7-s − 8-s − 9-s + 3.51i·10-s + 6.05i·11-s + i·12-s + 3.78·13-s + i·14-s + 3.51·15-s + 16-s + (−2.37 − 3.37i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 1.57i·5-s − 0.408i·6-s − 0.377i·7-s − 0.353·8-s − 0.333·9-s + 1.11i·10-s + 1.82i·11-s + 0.288i·12-s + 1.04·13-s + 0.267i·14-s + 0.906·15-s + 0.250·16-s + (−0.575 − 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928893 - 0.482264i\)
\(L(\frac12)\) \(\approx\) \(0.928893 - 0.482264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
7 \( 1 + iT \)
17 \( 1 + (2.37 + 3.37i)T \)
good5 \( 1 + 3.51iT - 5T^{2} \)
11 \( 1 - 6.05iT - 11T^{2} \)
13 \( 1 - 3.78T + 13T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + 6.74iT - 23T^{2} \)
29 \( 1 - 1.72iT - 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 + 5.78iT - 37T^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 7.02T + 47T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 - 7.84iT - 61T^{2} \)
67 \( 1 - 3.23T + 67T^{2} \)
71 \( 1 + 3.18iT - 71T^{2} \)
73 \( 1 + 11.0iT - 73T^{2} \)
79 \( 1 - 7.78iT - 79T^{2} \)
83 \( 1 + 6.33T + 83T^{2} \)
89 \( 1 + 8.53T + 89T^{2} \)
97 \( 1 - 18.0iT - 97T^{2} \)
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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

−10.08815862346625588541611492765, −9.231901669618571514979491855522, −8.943921466943820806222120434787, −7.83034929497955841005646294486, −7.03516775642263473389889127105, −5.67456658837855992520538948259, −4.72078099750313206483712180818, −4.01640554798097323201334911059, −2.18420836734017886982857441711, −0.76714636267887904046489409571, 1.37874990499634031071927719431, 2.95496570997215110575952277786, 3.43078685096823278648402114678, 5.75249387414389591170828984997, 6.22872703441885118433889859408, 7.06728134249200936902659287351, 8.038291324814567552032412053330, 8.649070567453153200504497465575, 9.719320296751652973335392709488, 10.75126883833175983812325685005

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