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Properties

Label 2-700-28.27-c1-0-48
Degree $2$
Conductor $700$
Sign $0.377 + 0.925i$
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 + 2.44·3-s + 2i·4-s + (−2.44 − 2.44i)6-s + (2.44 − i)7-s + (2 − 2i)8-s + 2.99·9-s − 5i·11-s + 4.89i·12-s − 2.44i·13-s + (−3.44 − 1.44i)14-s − 4·16-s + 4.89i·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 + 1.41·3-s + i·4-s + (−0.999 − 0.999i)6-s + (0.925 − 0.377i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s − 1.50i·11-s + 1.41i·12-s − 0.679i·13-s + (−0.921 − 0.387i)14-s − 16-s + 1.18i·17-s + (−0.707 − 0.707i)18-s + (1.30 − 0.534i)21-s + (−1.06 + 1.06i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.377 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52778 - 1.02648i\)
\(L(\frac12)\) \(\approx\) \(1.52778 - 1.02648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + i)T \)
good3 \( 1 - 2.44T + 3T^{2} \)
11 \( 1 + 5iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 12.2iT - 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 2.44iT - 73T^{2} \)
79 \( 1 - 9iT - 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 - 2.44iT - 89T^{2} \)
97 \( 1 + 7.34iT - 97T^{2} \)
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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

−10.32741162293571294656451986857, −9.241039230888776686960475290878, −8.474402430746762426551437648195, −8.119623082210598393738418763077, −7.37878809346440969740632087470, −5.83945417483856083674901699063, −4.20371664487996929524821622331, −3.42283279392700061331238687174, −2.46904096054098927735191071718, −1.19474953885959037910513850180, 1.72091051330494252790410592561, 2.54275857874761757624658307465, 4.34351713877144888435180303206, 5.07786503218918670005608996312, 6.54963878103986925820443929845, 7.53466264477991861895961651012, 7.904664301065001530583403273708, 8.975180802607994032299164299866, 9.359085594374512672837793948348, 10.15816113689060383096689646084

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