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Properties

Label 2-700-140.23-c1-0-15
Degree $2$
Conductor $700$
Sign $0.361 - 0.932i$
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  + (−1.41 + 0.0366i)2-s + (−0.402 − 0.107i)3-s + (1.99 − 0.103i)4-s + (0.572 + 0.137i)6-s + (−2.30 − 1.30i)7-s + (−2.81 + 0.219i)8-s + (−2.44 − 1.41i)9-s + (−0.725 + 0.418i)11-s + (−0.814 − 0.173i)12-s + (1.16 + 1.16i)13-s + (3.30 + 1.75i)14-s + (3.97 − 0.414i)16-s + (4.94 + 1.32i)17-s + (3.51 + 1.90i)18-s + (−2.91 + 5.05i)19-s + ⋯
L(s)  = 1  + (−0.999 + 0.0259i)2-s + (−0.232 − 0.0622i)3-s + (0.998 − 0.0518i)4-s + (0.233 + 0.0561i)6-s + (−0.870 − 0.492i)7-s + (−0.996 + 0.0777i)8-s + (−0.815 − 0.471i)9-s + (−0.218 + 0.126i)11-s + (−0.235 − 0.0500i)12-s + (0.322 + 0.322i)13-s + (0.882 + 0.469i)14-s + (0.994 − 0.103i)16-s + (1.20 + 0.321i)17-s + (0.827 + 0.449i)18-s + (−0.669 + 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.361 - 0.932i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460284 + 0.315192i\)
\(L(\frac12)\) \(\approx\) \(0.460284 + 0.315192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0366i)T \)
5 \( 1 \)
7 \( 1 + (2.30 + 1.30i)T \)
good3 \( 1 + (0.402 + 0.107i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.725 - 0.418i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.16 - 1.16i)T + 13iT^{2} \)
17 \( 1 + (-4.94 - 1.32i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.91 - 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.896 - 3.34i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.00iT - 29T^{2} \)
31 \( 1 + (-7.03 + 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.190 - 0.711i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.0958T + 41T^{2} \)
43 \( 1 + (-4.87 + 4.87i)T - 43iT^{2} \)
47 \( 1 + (5.28 - 1.41i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.43 - 12.8i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.46 - 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.919 + 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.138 + 0.515i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (1.55 - 5.78i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.30 + 9.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.36 - 4.36i)T - 83iT^{2} \)
89 \( 1 + (2.50 + 1.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.24 - 4.24i)T - 97iT^{2} \)
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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

−10.39312241948721132014494261277, −9.830245966262264707261128081251, −8.934514388370565717086001962950, −8.091213199843141276749561062450, −7.20839783674057730114476214868, −6.23770672994799616822920432111, −5.69157150061691442898086495354, −3.81529022387225281998270177085, −2.85743950348618944168848959120, −1.16852920541385247002268016859, 0.46831196184671230553128035661, 2.46388799109032684786511321573, 3.21578006964517828360678387343, 5.06764357718000528294941293574, 6.07150300700641316595875161479, 6.71072577923359149587425064992, 7.974990754852298118627995779747, 8.530634016195537434590741630371, 9.467216946636177522903143037091, 10.19662788385055523322934115226

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