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Properties

Label 2-700-140.19-c1-0-45
Degree $2$
Conductor $700$
Sign $0.789 + 0.614i$
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  + (0.334 + 1.37i)2-s + (−0.963 + 0.556i)3-s + (−1.77 + 0.919i)4-s + (−1.08 − 1.13i)6-s + (−1.26 − 2.32i)7-s + (−1.85 − 2.13i)8-s + (−0.880 + 1.52i)9-s + (−1.48 + 0.856i)11-s + (1.20 − 1.87i)12-s + 2.45·13-s + (2.77 − 2.51i)14-s + (2.30 − 3.26i)16-s + (−3.10 − 5.38i)17-s + (−2.39 − 0.699i)18-s + (−0.108 + 0.187i)19-s + ⋯
L(s)  = 1  + (0.236 + 0.971i)2-s + (−0.556 + 0.321i)3-s + (−0.888 + 0.459i)4-s + (−0.443 − 0.464i)6-s + (−0.477 − 0.878i)7-s + (−0.656 − 0.753i)8-s + (−0.293 + 0.508i)9-s + (−0.447 + 0.258i)11-s + (0.346 − 0.541i)12-s + 0.682·13-s + (0.740 − 0.672i)14-s + (0.577 − 0.816i)16-s + (−0.754 − 1.30i)17-s + (−0.563 − 0.164i)18-s + (−0.0248 + 0.0430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.789 + 0.614i$
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: \((2,\ 700,\ (\ :1/2),\ 0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525578 - 0.180488i\)
\(L(\frac12)\) \(\approx\) \(0.525578 - 0.180488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.334 - 1.37i)T \)
5 \( 1 \)
7 \( 1 + (1.26 + 2.32i)T \)
good3 \( 1 + (0.963 - 0.556i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.48 - 0.856i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.45T + 13T^{2} \)
17 \( 1 + (3.10 + 5.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.108 - 0.187i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.28 + 5.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.47T + 29T^{2} \)
31 \( 1 + (0.0819 + 0.141i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.66 + 3.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (-10.1 - 5.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.2 - 6.51i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 + 3.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.58 + 4.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.04iT - 71T^{2} \)
73 \( 1 + (3.80 + 6.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.8 + 7.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.47iT - 83T^{2} \)
89 \( 1 + (-1.54 - 0.891i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.46522540268734856650343976293, −9.366627727722744312608610704667, −8.562450748165650350137362710628, −7.49586709956144463560194075019, −6.82942060362526651088968013668, −5.90396397350794226711279256783, −4.91294124927743263984731142104, −4.25935289730491875255866971111, −2.90992581673395980628310404651, −0.30040099526331380388538522592, 1.45895676799478967564129067289, 2.87994549869289974918646775712, 3.78102662591744307509975462427, 5.20182225936571226763101377127, 5.91178729630141248734517978533, 6.66842412595691916593743417540, 8.363082164444234604064543872557, 8.874571960338881723143663705194, 9.822621139494596176563650625900, 10.76657501982095940471138370996

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