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Properties

Label 2-714-51.44-c1-0-22
Degree $2$
Conductor $714$
Sign $0.561 + 0.827i$
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  + (−0.923 − 0.382i)2-s + (0.285 + 1.70i)3-s + (0.707 + 0.707i)4-s + (−0.344 − 1.73i)5-s + (0.390 − 1.68i)6-s + (−0.980 − 0.195i)7-s + (−0.382 − 0.923i)8-s + (−2.83 + 0.974i)9-s + (−0.344 + 1.73i)10-s + (0.224 − 0.150i)11-s + (−1.00 + 1.40i)12-s + (2.13 − 2.13i)13-s + (0.831 + 0.555i)14-s + (2.85 − 1.08i)15-s + i·16-s + (−0.339 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (0.164 + 0.986i)3-s + (0.353 + 0.353i)4-s + (−0.154 − 0.774i)5-s + (0.159 − 0.688i)6-s + (−0.370 − 0.0737i)7-s + (−0.135 − 0.326i)8-s + (−0.945 + 0.324i)9-s + (−0.108 + 0.547i)10-s + (0.0677 − 0.0452i)11-s + (−0.290 + 0.406i)12-s + (0.592 − 0.592i)13-s + (0.222 + 0.148i)14-s + (0.738 − 0.279i)15-s + 0.250i·16-s + (−0.0822 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817217 - 0.433036i\)
\(L(\frac12)\) \(\approx\) \(0.817217 - 0.433036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
3 \( 1 + (-0.285 - 1.70i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (0.339 + 4.10i)T \)
good5 \( 1 + (0.344 + 1.73i)T + (-4.61 + 1.91i)T^{2} \)
11 \( 1 + (-0.224 + 0.150i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-2.13 + 2.13i)T - 13iT^{2} \)
19 \( 1 + (-0.806 + 1.94i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.20 + 1.80i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-5.50 + 1.09i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (2.23 - 3.33i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (2.55 + 1.70i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.65 + 8.32i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.20 + 5.32i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (-6.17 - 6.17i)T + 47iT^{2} \)
53 \( 1 + (-5.21 - 2.16i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.32 - 8.02i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.38 + 11.9i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + 4.91iT - 67T^{2} \)
71 \( 1 + (0.565 - 0.846i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-5.02 + 1.00i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (2.43 + 3.65i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-3.99 + 9.64i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.15 - 2.15i)T - 89iT^{2} \)
97 \( 1 + (0.490 + 2.46i)T + (-89.6 + 37.1i)T^{2} \)
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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

−10.32782241950872481306311580119, −9.266676367700636143762953632251, −8.860760174879020272203105242922, −8.062603533657386695638661424248, −6.91612273894558152006064433474, −5.64654354396941269928536163765, −4.71430500247699937238087323101, −3.65087897056566415993316735238, −2.61270514160345262913679453693, −0.63030993484744813844608906819, 1.38319305250304285065072978123, 2.64449568489957357548269526951, 3.79228085206439266172284963346, 5.63272960786170006889842988933, 6.50775562160917612019760027877, 6.96082643173734685819663761872, 7.999692566014884289413446098868, 8.610011925821924871267571690173, 9.604626528930793941590382713800, 10.54589853630550161991358938281

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