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Properties

Label 2-700-7.2-c1-0-11
Degree $2$
Conductor $700$
Sign $-0.266 + 0.963i$
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 + (−2.5 − 0.866i)7-s + (1 − 1.73i)9-s + (−3 − 5.19i)11-s − 2·13-s + (−3 − 5.19i)17-s + (−4 + 6.92i)19-s + (−0.500 − 2.59i)21-s + (1.5 − 2.59i)23-s + 5·27-s + 3·29-s + (−1 − 1.73i)31-s + (3 − 5.19i)33-s + (4 − 6.92i)37-s + (−1 − 1.73i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.944 − 0.327i)7-s + (0.333 − 0.577i)9-s + (−0.904 − 1.56i)11-s − 0.554·13-s + (−0.727 − 1.26i)17-s + (−0.917 + 1.58i)19-s + (−0.109 − 0.566i)21-s + (0.312 − 0.541i)23-s + 0.962·27-s + 0.557·29-s + (−0.179 − 0.311i)31-s + (0.522 − 0.904i)33-s + (0.657 − 1.13i)37-s + (−0.160 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518860 - 0.682032i\)
\(L(\frac12)\) \(\approx\) \(0.518860 - 0.682032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
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 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.25953784270951621519398279311, −9.337261374038248347063379468643, −8.658489491408132230019815303291, −7.62258246659032007455583264361, −6.57626711563376626277339237689, −5.79519194074867343360261129083, −4.51969856602132820936361931975, −3.51753398569155822286679401121, −2.67440762846505507262304340718, −0.40420496194072384370824523304, 2.00191864766087194181105551217, 2.74985629227441079877287027896, 4.37290623962142660537668212436, 5.16274023227331170122833576164, 6.65433293208522760385499514295, 7.02946516169490005460335523496, 8.087215895227938877152555156762, 8.918974111919296854243435642879, 10.00324023499322021332252933782, 10.41362188078514003608359287354

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