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Properties

Label 2-700-28.3-c1-0-32
Degree $2$
Conductor $700$
Sign $0.506 + 0.862i$
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  + (−0.288 − 1.38i)2-s + (0.450 − 0.780i)3-s + (−1.83 + 0.798i)4-s + (−1.21 − 0.398i)6-s + (2.29 − 1.30i)7-s + (1.63 + 2.30i)8-s + (1.09 + 1.89i)9-s + (3.24 + 1.87i)11-s + (−0.202 + 1.79i)12-s + 2.41i·13-s + (−2.47 − 2.80i)14-s + (2.72 − 2.92i)16-s + (0.505 + 0.291i)17-s + (2.30 − 2.06i)18-s + (3.07 + 5.33i)19-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)2-s + (0.260 − 0.450i)3-s + (−0.916 + 0.399i)4-s + (−0.494 − 0.162i)6-s + (0.869 − 0.494i)7-s + (0.578 + 0.815i)8-s + (0.364 + 0.631i)9-s + (0.977 + 0.564i)11-s + (−0.0585 + 0.517i)12-s + 0.671i·13-s + (−0.661 − 0.750i)14-s + (0.680 − 0.732i)16-s + (0.122 + 0.0707i)17-s + (0.543 − 0.485i)18-s + (0.706 + 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.506 + 0.862i$
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: \((2,\ 700,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40383 - 0.803268i\)
\(L(\frac12)\) \(\approx\) \(1.40383 - 0.803268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.288 + 1.38i)T \)
5 \( 1 \)
7 \( 1 + (-2.29 + 1.30i)T \)
good3 \( 1 + (-0.450 + 0.780i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.24 - 1.87i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.41iT - 13T^{2} \)
17 \( 1 + (-0.505 - 0.291i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.07 - 5.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.435T + 29T^{2} \)
31 \( 1 + (-1.26 + 2.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.65 + 9.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.35iT - 41T^{2} \)
43 \( 1 + 5.80iT - 43T^{2} \)
47 \( 1 + (-5.78 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.55 - 2.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.73 + 3.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.99 - 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.52 - 4.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.96iT - 71T^{2} \)
73 \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.397 + 0.229i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + (8.55 - 4.94i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.54iT - 97T^{2} \)
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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

−10.39295363427268366571405206908, −9.552315658691274286267401492955, −8.685078148088543470546396724485, −7.70458216584139071539264597139, −7.24505093422419116895140097668, −5.61994972933546055765557827627, −4.41969111333366381119231629663, −3.78140005486574857981845541544, −2.06334771878622122453001655337, −1.42747652266068465734193393270, 1.13199219369193030973908726704, 3.21440342933127530425380718191, 4.37439883188040736035252330151, 5.18473240947939256646899729109, 6.23258650267539193696138988422, 7.03984878064138429624432859154, 8.193359014806216574932042231883, 8.724924930497671790525700281345, 9.517320483380932721221831261726, 10.26868032315461018657650697784

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