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Properties

Label 2-704-11.4-c1-0-21
Degree $2$
Conductor $704$
Sign $-0.995 + 0.0913i$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
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  + (0.809 − 2.48i)3-s + (−1.30 + 0.951i)5-s + (−1.19 − 3.66i)7-s + (−3.11 − 2.26i)9-s + (−1.23 − 3.07i)11-s + (1.92 + 1.40i)13-s + (1.30 + 4.02i)15-s + (−1.92 + 1.40i)17-s + (−1.19 + 3.66i)19-s − 10.0·21-s − 2.47·23-s + (−0.736 + 2.26i)25-s + (−1.80 + 1.31i)27-s + (−2.66 − 8.19i)29-s + (−0.690 − 0.502i)31-s + ⋯
L(s)  = 1  + (0.467 − 1.43i)3-s + (−0.585 + 0.425i)5-s + (−0.450 − 1.38i)7-s + (−1.03 − 0.755i)9-s + (−0.372 − 0.927i)11-s + (0.534 + 0.388i)13-s + (0.337 + 1.04i)15-s + (−0.467 + 0.339i)17-s + (−0.273 + 0.840i)19-s − 2.20·21-s − 0.515·23-s + (−0.147 + 0.453i)25-s + (−0.348 + 0.252i)27-s + (−0.494 − 1.52i)29-s + (−0.124 − 0.0901i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.995 + 0.0913i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -0.995 + 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0473962 - 1.03504i\)
\(L(\frac12)\) \(\approx\) \(0.0473962 - 1.03504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (1.23 + 3.07i)T \)
good3 \( 1 + (-0.809 + 2.48i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.92 - 1.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.92 - 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.19 - 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (2.66 + 8.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.690 + 0.502i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.572 - 1.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.66 + 8.19i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.427 + 1.31i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.30 + 2.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.336 + 1.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.92 + 1.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (-5.16 + 3.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.281 + 0.865i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.78 - 4.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.5 + 7.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 + (-11.7 - 8.55i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.17656533869830129939218852570, −8.895505725857708808581226935224, −7.894500091705615100523912387927, −7.58535378819339680378677560041, −6.61890992781271299429149767857, −5.99498783673428524466660878488, −4.06462190240710091871075370879, −3.36953520543249603467193659566, −1.93867406905306822173357077615, −0.49507403361274383421885702279, 2.42653311690763419936965979003, 3.39753451073728063607618607773, 4.51484966660585505930634030663, 5.11738289459668878902935492869, 6.24974107636490625514315421361, 7.62698031153471225397865293368, 8.718538220330540260549534675775, 9.038690813895699318240555330347, 9.856360243350602421701272543296, 10.70988623612479747647037133077

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