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Properties

Label 2-700-35.4-c1-0-6
Degree $2$
Conductor $700$
Sign $0.897 + 0.441i$
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  + (−0.866 − 0.5i)3-s + (1.73 + 2i)7-s + (−1 − 1.73i)9-s + (1.5 − 2.59i)11-s + 2i·13-s + (2.59 + 1.5i)17-s + (−0.5 − 0.866i)19-s + (−0.499 − 2.59i)21-s + (2.59 − 1.5i)23-s + 5i·27-s + 6·29-s + (3.5 − 6.06i)31-s + (−2.59 + 1.5i)33-s + (0.866 − 0.5i)37-s + (1 − 1.73i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.654 + 0.755i)7-s + (−0.333 − 0.577i)9-s + (0.452 − 0.783i)11-s + 0.554i·13-s + (0.630 + 0.363i)17-s + (−0.114 − 0.198i)19-s + (−0.109 − 0.566i)21-s + (0.541 − 0.312i)23-s + 0.962i·27-s + 1.11·29-s + (0.628 − 1.08i)31-s + (−0.452 + 0.261i)33-s + (0.142 − 0.0821i)37-s + (0.160 − 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34322 - 0.312241i\)
\(L(\frac12)\) \(\approx\) \(1.34322 - 0.312241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-2.59 - 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-7.79 + 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.59 + 1.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.58228229690028579587625957511, −9.330331143770651479197706482952, −8.730194098665843714287815495433, −7.86366889247894960880346816183, −6.61471578461042561160903581751, −6.00075083894100920395407015869, −5.10325235297456487298999083648, −3.86496266686828701105150565772, −2.54908013316194272179441710969, −0.995514160080586106266598406927, 1.20192676082104167361428121210, 2.83416821354201037765539083348, 4.29848960580906198104923715898, 4.94392405431303223645198880202, 5.92711842536935524451389851746, 7.11819984222519069098002794999, 7.81523461678017251021654824563, 8.744523783228056246188299062794, 9.966506999469627039730863512299, 10.46316978812784146183132950551

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