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Properties

Label 2-700-140.139-c1-0-54
Degree $2$
Conductor $700$
Sign $0.465 + 0.884i$
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  + (1.39 + 0.207i)2-s − 1.47i·3-s + (1.91 + 0.579i)4-s + (0.305 − 2.06i)6-s + (−0.819 − 2.51i)7-s + (2.55 + 1.20i)8-s + 0.828·9-s − 2.79i·11-s + (0.853 − 2.82i)12-s − 5.83·13-s + (−0.625 − 3.68i)14-s + (3.32 + 2.21i)16-s + 4.12·17-s + (1.15 + 0.171i)18-s + 5.64·19-s + ⋯
L(s)  = 1  + (0.989 + 0.146i)2-s − 0.850i·3-s + (0.957 + 0.289i)4-s + (0.124 − 0.841i)6-s + (−0.309 − 0.950i)7-s + (0.904 + 0.426i)8-s + 0.276·9-s − 0.843i·11-s + (0.246 − 0.814i)12-s − 1.61·13-s + (−0.167 − 0.985i)14-s + (0.832 + 0.554i)16-s + 0.999·17-s + (0.273 + 0.0404i)18-s + 1.29·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.465 + 0.884i$
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: \((2,\ 700,\ (\ :1/2),\ 0.465 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38175 - 1.43771i\)
\(L(\frac12)\) \(\approx\) \(2.38175 - 1.43771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.207i)T \)
5 \( 1 \)
7 \( 1 + (0.819 + 2.51i)T \)
good3 \( 1 + 1.47iT - 3T^{2} \)
11 \( 1 + 2.79iT - 11T^{2} \)
13 \( 1 + 5.83T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 5.64T + 19T^{2} \)
23 \( 1 + 3.95T + 23T^{2} \)
29 \( 1 - 0.242T + 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 - 4.12iT - 41T^{2} \)
43 \( 1 - 5.59T + 43T^{2} \)
47 \( 1 - 6.25iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 2.94T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 - 7.43T + 67T^{2} \)
71 \( 1 - 7.23iT - 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 7.91iT - 79T^{2} \)
83 \( 1 + 1.47iT - 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 - 8.24T + 97T^{2} \)
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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

−10.29863470305571557139097256628, −9.751082272221029144230608230177, −7.977014332433320677259852136226, −7.50199763922747728825312904609, −6.82878766562223915364395444821, −5.85288952868668971410462897858, −4.85082106395993537378852247995, −3.72210718877009363891774466550, −2.68880666049150549261804401125, −1.17538802946832541639135451976, 2.04626337067381079835487379965, 3.13505712984721936705027407276, 4.20466202196102214528873907998, 5.14057143776003852881295650796, 5.66984622611191455290654025678, 7.07807579143394954010456502571, 7.66794631836289873998431705104, 9.372523172075278859977103700464, 9.794549153060273143233897558246, 10.49029991750555403264526131042

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