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Properties

Label 2-700-25.6-c1-0-4
Degree $2$
Conductor $700$
Sign $0.859 - 0.510i$
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  + (0.918 − 0.667i)3-s + (1.45 + 1.69i)5-s − 7-s + (−0.529 + 1.62i)9-s + (1.20 + 3.69i)11-s + (2.04 − 6.29i)13-s + (2.46 + 0.590i)15-s + (5.28 + 3.83i)17-s + (−4.18 − 3.04i)19-s + (−0.918 + 0.667i)21-s + (2.42 + 7.46i)23-s + (−0.773 + 4.93i)25-s + (1.65 + 5.08i)27-s + (−2.09 + 1.52i)29-s + (1.62 + 1.18i)31-s + ⋯
L(s)  = 1  + (0.530 − 0.385i)3-s + (0.650 + 0.759i)5-s − 0.377·7-s + (−0.176 + 0.542i)9-s + (0.361 + 1.11i)11-s + (0.566 − 1.74i)13-s + (0.637 + 0.152i)15-s + (1.28 + 0.931i)17-s + (−0.961 − 0.698i)19-s + (−0.200 + 0.145i)21-s + (0.505 + 1.55i)23-s + (−0.154 + 0.987i)25-s + (0.318 + 0.978i)27-s + (−0.389 + 0.282i)29-s + (0.292 + 0.212i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.859 - 0.510i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.859 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90280 + 0.522360i\)
\(L(\frac12)\) \(\approx\) \(1.90280 + 0.522360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 1.69i)T \)
7 \( 1 + T \)
good3 \( 1 + (-0.918 + 0.667i)T + (0.927 - 2.85i)T^{2} \)
11 \( 1 + (-1.20 - 3.69i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.04 + 6.29i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-5.28 - 3.83i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.18 + 3.04i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.42 - 7.46i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.09 - 1.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.62 - 1.18i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.07 + 9.47i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.63 + 11.1i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.67T + 43T^{2} \)
47 \( 1 + (3.45 - 2.50i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.36 - 1.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.52 - 4.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.39 + 4.30i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.25 + 2.36i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.00 + 2.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.74 - 5.36i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.87 + 2.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.15 + 4.47i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.78 + 11.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.7 - 7.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.54353497406343416276658546770, −9.718745836286231422499906030506, −8.825297614911601913829686035052, −7.69102875872475534538418883515, −7.26670921323493141544788204220, −6.01574632316256472432993072801, −5.35572278253030110481494883044, −3.68314651899629181394577003195, −2.78154694656414565623777669229, −1.65765197615965194825210872276, 1.10982005674622822677055605210, 2.71621599583117897650417848977, 3.83859249448815520146041819706, 4.73817080586559436620191990414, 6.18789920617288141975966568785, 6.43702406815147416395969940987, 8.181103495145061490439170023540, 8.762020163324762056655521048705, 9.455627768778157848704269002920, 10.02698556487725441469336642665

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