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Properties

Label 2-714-119.100-c1-0-10
Degree $2$
Conductor $714$
Sign $0.979 + 0.203i$
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  + (−0.965 + 0.258i)2-s + (−0.608 + 0.793i)3-s + (0.866 − 0.499i)4-s + (0.459 − 3.48i)5-s + (0.382 − 0.923i)6-s + (1.51 + 2.16i)7-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.459 + 3.48i)10-s + (0.377 − 0.0497i)11-s + (−0.130 + 0.991i)12-s + 3.95i·13-s + (−2.02 − 1.70i)14-s + (2.48 + 2.48i)15-s + (0.500 − 0.866i)16-s + (1.64 − 3.78i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.351 + 0.458i)3-s + (0.433 − 0.249i)4-s + (0.205 − 1.56i)5-s + (0.156 − 0.377i)6-s + (0.573 + 0.819i)7-s + (−0.249 + 0.249i)8-s + (−0.0862 − 0.321i)9-s + (0.145 + 1.10i)10-s + (0.113 − 0.0149i)11-s + (−0.0376 + 0.286i)12-s + 1.09i·13-s + (−0.541 − 0.454i)14-s + (0.642 + 0.642i)15-s + (0.125 − 0.216i)16-s + (0.398 − 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08280 - 0.111421i\)
\(L(\frac12)\) \(\approx\) \(1.08280 - 0.111421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (-1.51 - 2.16i)T \)
17 \( 1 + (-1.64 + 3.78i)T \)
good5 \( 1 + (-0.459 + 3.48i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (-0.377 + 0.0497i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 - 3.95iT - 13T^{2} \)
19 \( 1 + (0.0537 - 0.0144i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.95 - 2.54i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (-6.19 + 2.56i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.77 + 8.83i)T + (-8.02 - 29.9i)T^{2} \)
37 \( 1 + (8.33 + 1.09i)T + (35.7 + 9.57i)T^{2} \)
41 \( 1 + (1.96 + 0.815i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.83 + 4.83i)T - 43iT^{2} \)
47 \( 1 + (-2.91 - 1.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.802 + 2.99i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-10.2 - 2.74i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.06 + 5.42i)T + (15.7 - 58.9i)T^{2} \)
67 \( 1 + (-7.77 - 13.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.889 + 2.14i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-10.9 - 8.42i)T + (18.8 + 70.5i)T^{2} \)
79 \( 1 + (2.07 + 2.69i)T + (-20.4 + 76.3i)T^{2} \)
83 \( 1 + (7.53 + 7.53i)T + 83iT^{2} \)
89 \( 1 + (-2.99 - 1.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.560 - 0.232i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.04450853157417639860766079041, −9.417304954239450491278630049735, −8.770893524850887770493085229876, −8.178738875052193523685917276289, −6.91912721261072262405315790228, −5.72277145564374024965298844819, −5.10374900515622628984099222020, −4.21784041097398635243924327815, −2.27633157515579188085364188013, −0.939791304462965816217833546335, 1.16795689056725922639128144644, 2.61529102740898894717327002385, 3.57834834462970370712337434394, 5.17375698537428158152258715512, 6.49785086090239255866343461127, 6.87490366477881158660202656934, 7.85138200610198861388018805492, 8.478398994724795161023582221302, 10.07432417952345288252481129710, 10.53942689056939300525412674715

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