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Properties

Label 2-700-35.9-c1-0-4
Degree $2$
Conductor $700$
Sign $0.850 - 0.525i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (−2.77 + 1.60i)3-s + (2.15 + 1.53i)7-s + (3.63 − 6.29i)9-s + (−2.10 − 3.64i)11-s + 0.204i·13-s + (4.38 − 2.53i)17-s + (0.531 − 0.921i)19-s + (−8.44 − 0.794i)21-s + (−1.85 − 1.07i)23-s + 13.6i·27-s + 7.47·29-s + (4.23 + 7.33i)31-s + (11.6 + 6.73i)33-s + (−9.19 − 5.30i)37-s + (−0.327 − 0.567i)39-s + ⋯
L(s)  = 1  + (−1.60 + 0.925i)3-s + (0.815 + 0.579i)7-s + (1.21 − 2.09i)9-s + (−0.633 − 1.09i)11-s + 0.0566i·13-s + (1.06 − 0.614i)17-s + (0.122 − 0.211i)19-s + (−1.84 − 0.173i)21-s + (−0.386 − 0.223i)23-s + 2.63i·27-s + 1.38·29-s + (0.760 + 1.31i)31-s + (2.03 + 1.17i)33-s + (−1.51 − 0.872i)37-s + (−0.0524 − 0.0908i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.896829 + 0.254502i\)
\(L(\frac12)\) \(\approx\) \(0.896829 + 0.254502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.15 - 1.53i)T \)
good3 \( 1 + (2.77 - 1.60i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.10 + 3.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.204iT - 13T^{2} \)
17 \( 1 + (-4.38 + 2.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.531 + 0.921i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.47T + 29T^{2} \)
31 \( 1 + (-4.23 - 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.19 + 5.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 8.26iT - 43T^{2} \)
47 \( 1 + (-2.83 - 1.63i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 + 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.602 - 1.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.827 - 1.43i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.591T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.27 + 5.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.88iT - 83T^{2} \)
89 \( 1 + (4.63 - 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.33iT - 97T^{2} \)
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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

−10.69792632521444680731168097703, −9.948610241588530757167839246510, −8.943378876389616256138889983466, −7.988765321906614659484240031551, −6.70731783432121352279211178170, −5.70430807019798661156152212967, −5.25593422425414356567797894468, −4.39348400585436746805387902225, −3.03500130909589357580016552661, −0.878352176366725318819736787854, 0.953925877521022935821534277266, 2.09540899472882166131207156095, 4.23622115381379075533158626603, 5.11647913886430763477492524714, 5.84676253365149981126152236244, 6.89665448181193146246852498554, 7.57632994789981256808367709109, 8.211026704076022510478199221146, 10.10986969000213640916569572569, 10.35367770301527274576847209034

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