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Properties

Label 2-700-140.19-c1-0-20
Degree $2$
Conductor $700$
Sign $-0.295 - 0.955i$
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  + (−0.468 + 1.33i)2-s + (−2.22 + 1.28i)3-s + (−1.56 − 1.25i)4-s + (−0.670 − 3.57i)6-s + (−2.62 + 0.304i)7-s + (2.40 − 1.49i)8-s + (1.80 − 3.12i)9-s + (5.44 − 3.14i)11-s + (5.08 + 0.780i)12-s + 3.00·13-s + (0.826 − 3.64i)14-s + (0.868 + 3.90i)16-s + (−0.539 − 0.935i)17-s + (3.32 + 3.87i)18-s + (−3.18 + 5.51i)19-s + ⋯
L(s)  = 1  + (−0.331 + 0.943i)2-s + (−1.28 + 0.742i)3-s + (−0.780 − 0.625i)4-s + (−0.273 − 1.45i)6-s + (−0.993 + 0.115i)7-s + (0.848 − 0.528i)8-s + (0.601 − 1.04i)9-s + (1.64 − 0.947i)11-s + (1.46 + 0.225i)12-s + 0.834·13-s + (0.220 − 0.975i)14-s + (0.217 + 0.976i)16-s + (−0.130 − 0.226i)17-s + (0.783 + 0.912i)18-s + (−0.730 + 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.295 - 0.955i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396636 + 0.537819i\)
\(L(\frac12)\) \(\approx\) \(0.396636 + 0.537819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.468 - 1.33i)T \)
5 \( 1 \)
7 \( 1 + (2.62 - 0.304i)T \)
good3 \( 1 + (2.22 - 1.28i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.44 + 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
17 \( 1 + (0.539 + 0.935i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.18 - 5.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 2.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.512T + 29T^{2} \)
31 \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.56 - 2.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 0.683T + 43T^{2} \)
47 \( 1 + (-9.71 - 5.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.93 - 1.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.62 + 2.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.28 - 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.78 + 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + (-1.26 - 2.18i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.18 - 5.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.37iT - 83T^{2} \)
89 \( 1 + (-7.79 - 4.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.56T + 97T^{2} \)
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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

−10.64911631323198560538427520220, −9.708142377453939569176863336922, −9.135483156487122536958410752362, −8.206337735671945982126138250399, −6.73357904286634671888854028898, −6.12288753026466646664465343864, −5.80804671596569617674231435328, −4.37561459652228905698947393760, −3.70886199362758771903807262254, −0.889789965706809016904258767057, 0.70989249187305126708385849353, 1.90691407675159734017895273604, 3.56554828434708390962747152738, 4.51603611707974031296371666201, 5.82193881184518556507818745461, 6.76366040026259504855878257730, 7.25873256396415601105125843732, 8.901805806946992526663359502150, 9.319377550027967171783220060053, 10.51197531577751842917895330165

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