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Properties

Label 2-700-140.139-c1-0-2
Degree $2$
Conductor $700$
Sign $-0.997 - 0.0759i$
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 − i)2-s + 2.44i·3-s + 2i·4-s + (2.44 − 2.44i)6-s + (−1 − 2.44i)7-s + (2 − 2i)8-s − 2.99·9-s + 5i·11-s − 4.89·12-s − 2.44·13-s + (−1.44 + 3.44i)14-s − 4·16-s − 4.89·17-s + (2.99 + 2.99i)18-s + (5.99 − 2.44i)21-s + (5 − 5i)22-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.41i·3-s + i·4-s + (0.999 − 0.999i)6-s + (−0.377 − 0.925i)7-s + (0.707 − 0.707i)8-s − 0.999·9-s + 1.50i·11-s − 1.41·12-s − 0.679·13-s + (−0.387 + 0.921i)14-s − 16-s − 1.18·17-s + (0.707 + 0.707i)18-s + (1.30 − 0.534i)21-s + (1.06 − 1.06i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.997 - 0.0759i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.997 - 0.0759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0109045 + 0.286631i\)
\(L(\frac12)\) \(\approx\) \(0.0109045 + 0.286631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + 4.89iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 - 3T + 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 + 2.44T + 73T^{2} \)
79 \( 1 - 9iT - 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 + 2.44iT - 89T^{2} \)
97 \( 1 - 7.34T + 97T^{2} \)
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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

−10.40839393795982131712603024176, −10.18352901677318683739359004870, −9.435025261206847219749978556981, −8.722779947751577834796515121811, −7.42641557209902586737057996713, −6.81720838421729738503968428847, −4.89857709992370256171778083501, −4.30397711727848389678941555694, −3.44616769979758323797794276976, −2.06685534072767990501392757166, 0.17987227981376109836873039345, 1.73819859360698334423726265906, 2.87589362058556018888713648461, 4.95334382961862985307434193233, 6.13500724509747292012278700101, 6.39548296638332241740208188831, 7.43682482600369517638195388443, 8.277333435776522927375896824777, 8.834606139585573120289184908447, 9.734717193456334910350482464931

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