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Properties

Label 2-700-100.23-c1-0-36
Degree $2$
Conductor $700$
Sign $0.161 - 0.986i$
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  + (−1.36 − 0.375i)2-s + (1.09 + 2.14i)3-s + (1.71 + 1.02i)4-s + (2.03 + 0.921i)5-s + (−0.683 − 3.33i)6-s + (0.707 + 0.707i)7-s + (−1.95 − 2.04i)8-s + (−1.63 + 2.24i)9-s + (−2.43 − 2.02i)10-s + (1.62 + 2.23i)11-s + (−0.319 + 4.79i)12-s + (−0.628 − 3.96i)13-s + (−0.698 − 1.22i)14-s + (0.250 + 5.36i)15-s + (1.90 + 3.51i)16-s + (5.14 + 2.62i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.265i)2-s + (0.630 + 1.23i)3-s + (0.858 + 0.512i)4-s + (0.911 + 0.412i)5-s + (−0.279 − 1.35i)6-s + (0.267 + 0.267i)7-s + (−0.692 − 0.721i)8-s + (−0.544 + 0.749i)9-s + (−0.768 − 0.639i)10-s + (0.489 + 0.673i)11-s + (−0.0921 + 1.38i)12-s + (−0.174 − 1.10i)13-s + (−0.186 − 0.328i)14-s + (0.0645 + 1.38i)15-s + (0.475 + 0.879i)16-s + (1.24 + 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.161 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16689 + 0.991819i\)
\(L(\frac12)\) \(\approx\) \(1.16689 + 0.991819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.375i)T \)
5 \( 1 + (-2.03 - 0.921i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.09 - 2.14i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-1.62 - 2.23i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.628 + 3.96i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-5.14 - 2.62i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.07 - 3.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.04 + 6.62i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (5.43 + 1.76i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.54 - 2.45i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.05 + 1.27i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (9.51 + 6.91i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.93 + 2.93i)T - 43iT^{2} \)
47 \( 1 + (3.51 - 1.79i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (12.2 - 6.23i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.247 + 0.179i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.17 - 1.58i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-6.72 + 13.2i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (5.97 + 1.94i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.28 - 0.520i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.654 + 2.01i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.83 + 3.98i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-9.01 - 12.4i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.17 - 8.18i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.48902444468934483999482776534, −9.685733071961478397962264108248, −9.283360404496648271862111103659, −8.297241600522341192392014850872, −7.48266318013985137964292595370, −6.22052184193776174084089355283, −5.25228532887354711568895861934, −3.77668543454658615751707586560, −2.94443476592078293147003409698, −1.74324215124559676678827825154, 1.16650618353281922610210543763, 1.84897313141502487984930023924, 3.12993062663462684264138382368, 5.15857075371229305523381816097, 6.13665907570436106568901952946, 7.02027791877765401825351947302, 7.62395843894101892609009939442, 8.512696353560569272319534001308, 9.373720925284997820191315941099, 9.732204698885949819060157679747

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