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Properties

Label 2-704-11.3-c1-0-1
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: $\chi_{704} (641, \cdot )$
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

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

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