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Properties

Label 2-714-119.81-c1-0-13
Degree $2$
Conductor $714$
Sign $0.456 + 0.889i$
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  + (0.866 − 0.5i)2-s + (−0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s + (−0.707 + 0.707i)6-s + (1.02 + 2.43i)7-s − 0.999i·8-s + (0.866 − 0.499i)9-s + (−0.258 − 0.965i)10-s + (−1.40 − 5.22i)11-s + (−0.258 + 0.965i)12-s + 3.82·13-s + (2.10 + 1.59i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (−4.12 + 0.0693i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.557 + 0.149i)3-s + (0.249 − 0.433i)4-s + (0.115 − 0.431i)5-s + (−0.288 + 0.288i)6-s + (0.387 + 0.921i)7-s − 0.353i·8-s + (0.288 − 0.166i)9-s + (−0.0818 − 0.305i)10-s + (−0.422 − 1.57i)11-s + (−0.0747 + 0.278i)12-s + 1.06·13-s + (0.563 + 0.427i)14-s + 0.258i·15-s + (−0.125 − 0.216i)16-s + (−0.999 + 0.0168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63047 - 0.996355i\)
\(L(\frac12)\) \(\approx\) \(1.63047 - 0.996355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-1.02 - 2.43i)T \)
17 \( 1 + (4.12 - 0.0693i)T \)
good5 \( 1 + (-0.258 + 0.965i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.40 + 5.22i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
19 \( 1 + (-3.46 + 2i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.16 - 1.91i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.12 + 6.12i)T + 29iT^{2} \)
31 \( 1 + (-1.53 + 0.410i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.34 + 8.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.94 + 2.94i)T - 41iT^{2} \)
43 \( 1 + 3.75iT - 43T^{2} \)
47 \( 1 + (-4.62 - 8.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.42 + 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.10 - 1.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.22 + 1.40i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.41 - 7.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.58 + 3.58i)T + 71iT^{2} \)
73 \( 1 + (4.89 - 1.31i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.79 - 1.55i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 + (-0.828 - 1.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.48 - 8.48i)T + 97iT^{2} \)
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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

−10.81421001254615996869417108732, −9.141265131565192427128233798003, −8.949221151206179089897798620639, −7.64363874106951177403555419887, −6.28468769157657203584131146040, −5.63304724919471256835450810625, −4.99922796459901360410784761438, −3.74893026083380541206645678171, −2.60669101161777246558910054028, −0.996614942499960636697752162216, 1.54755760106310625322695031947, 3.11970501902739375802464857248, 4.44047058670285849992853263295, 4.97358037465073445677514836363, 6.25673242251235393050572327410, 7.03392135435986837912489910788, 7.52005345366333221651670626557, 8.730977369032780241002866653103, 9.954624118271204488223389926546, 10.80516159979848051856227762812

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