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Properties

Label 2-700-28.3-c1-0-26
Degree $2$
Conductor $700$
Sign $0.0734 - 0.997i$
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  + (1.24 + 0.674i)2-s + (0.366 − 0.634i)3-s + (1.09 + 1.67i)4-s + (0.883 − 0.542i)6-s + (−0.664 + 2.56i)7-s + (0.226 + 2.81i)8-s + (1.23 + 2.13i)9-s + (−2.33 − 1.34i)11-s + (1.46 − 0.0783i)12-s + 3.95i·13-s + (−2.55 + 2.73i)14-s + (−1.61 + 3.65i)16-s + (−1.22 − 0.709i)17-s + (0.0930 + 3.48i)18-s + (−1.61 − 2.79i)19-s + ⋯
L(s)  = 1  + (0.879 + 0.476i)2-s + (0.211 − 0.366i)3-s + (0.545 + 0.838i)4-s + (0.360 − 0.221i)6-s + (−0.250 + 0.967i)7-s + (0.0800 + 0.996i)8-s + (0.410 + 0.710i)9-s + (−0.702 − 0.405i)11-s + (0.422 − 0.0226i)12-s + 1.09i·13-s + (−0.682 + 0.731i)14-s + (−0.404 + 0.914i)16-s + (−0.298 − 0.172i)17-s + (0.0219 + 0.820i)18-s + (−0.369 − 0.640i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.0734 - 0.997i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.0734 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87168 + 1.73895i\)
\(L(\frac12)\) \(\approx\) \(1.87168 + 1.73895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.674i)T \)
5 \( 1 \)
7 \( 1 + (0.664 - 2.56i)T \)
good3 \( 1 + (-0.366 + 0.634i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.33 + 1.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.95iT - 13T^{2} \)
17 \( 1 + (1.22 + 0.709i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 + 2.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.25 + 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.17T + 29T^{2} \)
31 \( 1 + (-3.81 + 6.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.325iT - 41T^{2} \)
43 \( 1 - 9.28iT - 43T^{2} \)
47 \( 1 + (3.28 + 5.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.807 - 1.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.81 + 6.61i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.3 + 7.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.62 - 1.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.4iT - 71T^{2} \)
73 \( 1 + (1.22 + 0.709i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 + (4.10 - 2.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.35iT - 97T^{2} \)
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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

−10.99479202332251204394757670141, −9.696170431286007972067659452576, −8.574117123086378215886524192772, −8.053807447690648250838594484013, −6.88058083327930774431556806740, −6.33840363704959084616614678939, −5.12495454608897468514667446867, −4.47556046159650287093623951263, −2.90611668518326411484163104873, −2.18871406418646147134523388786, 1.04194310484974332678222373354, 2.78947495280499178516885278509, 3.68309783974697076711897969936, 4.53221064060746254735857129218, 5.51742739510568749742651532253, 6.65871671203049272643789340952, 7.36497603776069708715176914000, 8.606514188456861499240502698836, 9.916110058678256482114447076844, 10.23744426154921410863228274854

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