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Properties

Label 2-700-28.3-c1-0-13
Degree $2$
Conductor $700$
Sign $0.994 - 0.106i$
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  + (0.397 − 1.35i)2-s + (−0.556 + 0.963i)3-s + (−1.68 − 1.07i)4-s + (1.08 + 1.13i)6-s + (−2.32 − 1.26i)7-s + (−2.13 + 1.85i)8-s + (0.880 + 1.52i)9-s + (1.48 + 0.856i)11-s + (1.97 − 1.02i)12-s + 2.45i·13-s + (−2.63 + 2.65i)14-s + (1.67 + 3.63i)16-s + (5.38 + 3.10i)17-s + (2.42 − 0.589i)18-s + (−0.108 − 0.187i)19-s + ⋯
L(s)  = 1  + (0.280 − 0.959i)2-s + (−0.321 + 0.556i)3-s + (−0.842 − 0.539i)4-s + (0.443 + 0.464i)6-s + (−0.878 − 0.477i)7-s + (−0.753 + 0.656i)8-s + (0.293 + 0.508i)9-s + (0.447 + 0.258i)11-s + (0.570 − 0.295i)12-s + 0.682i·13-s + (−0.705 + 0.708i)14-s + (0.418 + 0.908i)16-s + (1.30 + 0.754i)17-s + (0.570 − 0.138i)18-s + (−0.0248 − 0.0430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.994 - 0.106i$
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: \((2,\ 700,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23719 + 0.0658172i\)
\(L(\frac12)\) \(\approx\) \(1.23719 + 0.0658172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.397 + 1.35i)T \)
5 \( 1 \)
7 \( 1 + (2.32 + 1.26i)T \)
good3 \( 1 + (0.556 - 0.963i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.48 - 0.856i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.45iT - 13T^{2} \)
17 \( 1 + (-5.38 - 3.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.108 + 0.187i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.68 + 3.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
31 \( 1 + (-0.0819 + 0.141i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.84 - 6.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.34iT - 41T^{2} \)
43 \( 1 - 1.89iT - 43T^{2} \)
47 \( 1 + (-5.85 - 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.51 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
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 \( 1 + (4.48 + 2.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.04iT - 71T^{2} \)
73 \( 1 + (6.59 + 3.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.8 - 7.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.47T + 83T^{2} \)
89 \( 1 + (1.54 - 0.891i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.44969795530818830857067218973, −9.827945193361549710365836697786, −9.227844993903476010662717693969, −8.007892511963995950851176805699, −6.76486549179248769710930611300, −5.76964824487436583532938571807, −4.66898539486003321998415163607, −3.97234369795929099900781732473, −2.91539227199489866923311377884, −1.32693137140157469890987492795, 0.72399192438142079271456235658, 3.03585424602657936768203383104, 3.92226642317027092728086596580, 5.60817160967029535757278266907, 5.75781628503234341141731081788, 7.13505431875915364610112162841, 7.29082886882047995230084331770, 8.715749173753059629506882223672, 9.342657686944132480997231064503, 10.17389676350916523495598228241

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