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Properties

Label 2-700-100.23-c1-0-42
Degree $2$
Conductor $700$
Sign $0.847 - 0.530i$
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  + (1.40 − 0.180i)2-s + (0.474 + 0.930i)3-s + (1.93 − 0.506i)4-s + (−2.11 + 0.712i)5-s + (0.832 + 1.21i)6-s + (−0.707 − 0.707i)7-s + (2.62 − 1.05i)8-s + (1.12 − 1.54i)9-s + (−2.84 + 1.38i)10-s + (2.90 + 4.00i)11-s + (1.38 + 1.56i)12-s + (0.496 + 3.13i)13-s + (−1.11 − 0.864i)14-s + (−1.66 − 1.63i)15-s + (3.48 − 1.95i)16-s + (4.14 + 2.11i)17-s + ⋯
L(s)  = 1  + (0.991 − 0.127i)2-s + (0.273 + 0.537i)3-s + (0.967 − 0.253i)4-s + (−0.947 + 0.318i)5-s + (0.340 + 0.497i)6-s + (−0.267 − 0.267i)7-s + (0.927 − 0.374i)8-s + (0.374 − 0.514i)9-s + (−0.899 + 0.436i)10-s + (0.876 + 1.20i)11-s + (0.400 + 0.450i)12-s + (0.137 + 0.869i)13-s + (−0.299 − 0.230i)14-s + (−0.430 − 0.422i)15-s + (0.871 − 0.489i)16-s + (1.00 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.847 - 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74037 + 0.786263i\)
\(L(\frac12)\) \(\approx\) \(2.74037 + 0.786263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.180i)T \)
5 \( 1 + (2.11 - 0.712i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.474 - 0.930i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-2.90 - 4.00i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.496 - 3.13i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.14 - 2.11i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.918 - 2.82i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.437 + 2.75i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (10.1 + 3.30i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.809 + 0.263i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.51 - 1.03i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (6.30 + 4.58i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.69 + 3.69i)T - 43iT^{2} \)
47 \( 1 + (-5.32 + 2.71i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (4.52 - 2.30i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (8.03 + 5.83i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.64 - 7.00i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.64 - 3.22i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-4.99 - 1.62i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.69 - 1.21i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.42 + 13.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.3 + 5.79i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (6.47 + 8.91i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.70 + 13.1i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.52291608596981218603903865863, −9.908388483123711754099601133545, −8.947971266503227608550733520477, −7.52628731791127458023813207627, −7.01712659333353985287489317454, −6.07711438686715737394761486679, −4.63417468929811793176358948794, −3.91315233366445545723159552968, −3.45014231076162085617215559444, −1.71719773798389403418200249908, 1.29532027658774376830456100069, 3.04718628443193430176353308780, 3.63801484244832679269814651905, 4.96043282989624144370239881767, 5.75771520168797803543040933675, 6.92044618694565418700972550049, 7.63866529558001470167395528332, 8.299946599546426118481163430776, 9.384441540314476418084147033073, 10.84611250302694881102864275060

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