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Properties

Label 2-700-140.19-c1-0-57
Degree $2$
Conductor $700$
Sign $0.535 + 0.844i$
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  + (−1.12 + 0.859i)2-s + (2.33 − 1.34i)3-s + (0.520 − 1.93i)4-s + (−1.45 + 3.51i)6-s + (0.0832 − 2.64i)7-s + (1.07 + 2.61i)8-s + (2.11 − 3.67i)9-s + (1.31 − 0.760i)11-s + (−1.38 − 5.20i)12-s + 1.14·13-s + (2.18 + 3.04i)14-s + (−3.45 − 2.01i)16-s + (−2.21 − 3.83i)17-s + (0.777 + 5.94i)18-s + (−3.04 + 5.28i)19-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (1.34 − 0.776i)3-s + (0.260 − 0.965i)4-s + (−0.595 + 1.43i)6-s + (0.0314 − 0.999i)7-s + (0.380 + 0.924i)8-s + (0.706 − 1.22i)9-s + (0.397 − 0.229i)11-s + (−0.399 − 1.50i)12-s + 0.316·13-s + (0.582 + 0.812i)14-s + (−0.864 − 0.502i)16-s + (−0.537 − 0.930i)17-s + (0.183 + 1.40i)18-s + (−0.699 + 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.535 + 0.844i$
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: \((2,\ 700,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40878 - 0.774436i\)
\(L(\frac12)\) \(\approx\) \(1.40878 - 0.774436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.859i)T \)
5 \( 1 \)
7 \( 1 + (-0.0832 + 2.64i)T \)
good3 \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.31 + 0.760i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.14T + 13T^{2} \)
17 \( 1 + (2.21 + 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.04 - 5.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.02 + 6.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + (-4.92 - 8.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.48 + 4.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.36iT - 41T^{2} \)
43 \( 1 + 4.89T + 43T^{2} \)
47 \( 1 + (-2.86 - 1.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.36 + 4.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.927 + 1.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.93 + 2.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.25 - 9.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.04iT - 71T^{2} \)
73 \( 1 + (3.93 + 6.80i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.54 - 2.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.98iT - 83T^{2} \)
89 \( 1 + (-12.9 - 7.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.20607299803344825756147777458, −8.980903514668220558506971129076, −8.605714511685237816836539239362, −7.79998337785265625914276632164, −6.92112167254356091082278078801, −6.51040740530518506433018516383, −4.85586934400075193113474817751, −3.52461334105009246009667308394, −2.20003686397210399013718854353, −0.992016576214118874098104326207, 1.88412913569230081235535229764, 2.80261596354106227943952146143, 3.71862197309864629992769855581, 4.71126516075837548124461415177, 6.35045096390864722012150482226, 7.51678830395209088585889304147, 8.605723955220754411918974767298, 8.786131284381806231247267622229, 9.555382191329138275785675967296, 10.31517884652915666751994510214

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