Properties

Label 2-700-7.4-c1-0-9
Degree $2$
Conductor $700$
Sign $0.605 + 0.795i$
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  + (0.5 − 0.866i)3-s + (2 − 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 2·13-s + (1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s + (−0.499 − 2.59i)21-s + (1.5 + 2.59i)23-s + 5·27-s − 6·29-s + (3.5 − 6.06i)31-s + (−1.5 − 2.59i)33-s + (−0.5 − 0.866i)37-s + (−1 + 1.73i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 0.554·13-s + (0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s + (−0.109 − 0.566i)21-s + (0.312 + 0.541i)23-s + 0.962·27-s − 1.11·29-s + (0.628 − 1.08i)31-s + (−0.261 − 0.452i)33-s + (−0.0821 − 0.142i)37-s + (−0.160 + 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66147 - 0.823639i\)
\(L(\frac12)\) \(\approx\) \(1.66147 - 0.823639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.35508363992139003417916379326, −9.458114382926935858904750426256, −8.427759257669920933694000067125, −7.59618411399666916661370408777, −7.15087059010579632517495786401, −5.81820520540124750831627349732, −4.82575660321742850830212789460, −3.76672952584091699620379785258, −2.38062798038959765339881574384, −1.10346059572545146215595998039, 1.59219404940058172339480430264, 2.94732356143077770675647655073, 4.21252357604103652490734770425, 4.92643690093785376021988999200, 6.10186939289148981230781709907, 7.12794565626510936920685402125, 8.068373590334878660937031328481, 9.045154973994390634652297060305, 9.563724693824625332199937660980, 10.48901972068012868972455417702

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