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Properties

Label 2-700-28.19-c1-0-53
Degree $2$
Conductor $700$
Sign $0.127 + 0.991i$
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  + (−0.365 + 1.36i)2-s + (−0.739 − 1.28i)3-s + (−1.73 − 0.998i)4-s + (2.02 − 0.542i)6-s + (2.56 − 0.664i)7-s + (1.99 − 2.00i)8-s + (0.406 − 0.703i)9-s + (−5.32 + 3.07i)11-s + (0.00185 + 2.95i)12-s − 3.33i·13-s + (−0.0291 + 3.74i)14-s + (2.00 + 3.46i)16-s + (2.20 − 1.27i)17-s + (0.812 + 0.811i)18-s + (−0.352 + 0.611i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.966i)2-s + (−0.426 − 0.739i)3-s + (−0.866 − 0.499i)4-s + (0.824 − 0.221i)6-s + (0.967 − 0.250i)7-s + (0.706 − 0.707i)8-s + (0.135 − 0.234i)9-s + (−1.60 + 0.927i)11-s + (0.000534 + 0.853i)12-s − 0.924i·13-s + (−0.00779 + 0.999i)14-s + (0.501 + 0.865i)16-s + (0.536 − 0.309i)17-s + (0.191 + 0.191i)18-s + (−0.0809 + 0.140i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.127 + 0.991i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550497 - 0.484396i\)
\(L(\frac12)\) \(\approx\) \(0.550497 - 0.484396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.365 - 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-2.56 + 0.664i)T \)
good3 \( 1 + (0.739 + 1.28i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (5.32 - 3.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
17 \( 1 + (-2.20 + 1.27i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.352 - 0.611i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.70 + 0.983i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.17T + 29T^{2} \)
31 \( 1 + (3.40 + 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.40 + 5.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.53iT - 41T^{2} \)
43 \( 1 + 4.59iT - 43T^{2} \)
47 \( 1 + (-2.18 + 3.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.77 + 4.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.40 + 5.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.07 - 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.51 - 1.45i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.37iT - 71T^{2} \)
73 \( 1 + (-2.20 + 1.27i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.38 + 3.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.70T + 83T^{2} \)
89 \( 1 + (-5.19 - 3.00i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.46iT - 97T^{2} \)
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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

−10.21484143505627836160033015862, −9.322598558090291938241343910739, −7.914374401733804168665476205302, −7.75448789660741441005829061406, −6.96689532503715047207485158006, −5.66096293698986401947341620032, −5.25072849191562433543382911694, −4.01379421119732954781678451757, −2.00953474643114207310552591528, −0.44898949171452496267332892760, 1.67076854007074716064317914315, 2.94166866355406867025771081161, 4.21699410091224705861219087230, 5.01589190412585580267849496042, 5.71260975925826522472132261572, 7.60081847337556365237178123671, 8.176920643292646545500539278753, 9.124079263197035818746007163915, 10.03170875537220870422626230593, 10.84282959404232809065815655591

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