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Properties

Label 2-714-357.230-c1-0-23
Degree $2$
Conductor $714$
Sign $0.962 - 0.271i$
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  + (0.707 + 0.707i)2-s + (−0.923 − 1.46i)3-s + 1.00i·4-s + (−0.224 + 0.541i)5-s + (0.382 − 1.68i)6-s + (1.81 − 1.92i)7-s + (−0.707 + 0.707i)8-s + (−1.29 + 2.70i)9-s + (−0.541 + 0.224i)10-s + (1.70 + 4.12i)11-s + (1.46 − 0.923i)12-s + 1.08·13-s + (2.64 − 0.0761i)14-s + (1 − 0.171i)15-s − 1.00·16-s + (3.76 − 1.68i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.533 − 0.845i)3-s + 0.500i·4-s + (−0.100 + 0.242i)5-s + (0.156 − 0.689i)6-s + (0.686 − 0.727i)7-s + (−0.250 + 0.250i)8-s + (−0.430 + 0.902i)9-s + (−0.171 + 0.0708i)10-s + (0.514 + 1.24i)11-s + (0.422 − 0.266i)12-s + 0.300·13-s + (0.706 − 0.0203i)14-s + (0.258 − 0.0442i)15-s − 0.250·16-s + (0.912 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79412 + 0.247940i\)
\(L(\frac12)\) \(\approx\) \(1.79412 + 0.247940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.923 + 1.46i)T \)
7 \( 1 + (-1.81 + 1.92i)T \)
17 \( 1 + (-3.76 + 1.68i)T \)
good5 \( 1 + (0.224 - 0.541i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-1.70 - 4.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.08T + 13T^{2} \)
19 \( 1 + (0.989 + 0.989i)T + 19iT^{2} \)
23 \( 1 + (2.53 + 6.12i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-9.24 - 3.82i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.38 + 5.76i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-6.12 - 2.53i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.21 - 2.93i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + 43iT^{2} \)
47 \( 1 - 9.55iT - 47T^{2} \)
53 \( 1 + (8 + 8i)T + 53iT^{2} \)
59 \( 1 + (4.01 + 4.01i)T + 59iT^{2} \)
61 \( 1 + (13.6 - 5.67i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 9.17T + 67T^{2} \)
71 \( 1 + (-0.778 + 1.87i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.83 - 3.24i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.82 - 3.65i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.54 + 5.54i)T - 83iT^{2} \)
89 \( 1 - 6.62iT - 89T^{2} \)
97 \( 1 + (12.3 + 5.09i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.69981883848503005265747683681, −9.664162971855064584899547847600, −8.249593348813920919883436124616, −7.69417061315530525820919087513, −6.82960767544777679538570790371, −6.28096909260076216011392791312, −4.93289323259794995607132562778, −4.37232428470402649156944604471, −2.74265625714385601492009890275, −1.24530101495343405543373165806, 1.14677165343202518669325828065, 2.95718127598089027704950267218, 3.92194654674323012831552604741, 4.87849065702838883010230395457, 5.76711392364240826933807010974, 6.30676778042990645777917088451, 8.109533179873630891059348560433, 8.778004036526728886447569107455, 9.681772224086192971825561382475, 10.62750392128764026809768816208

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