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Properties

Label 2-704-44.19-c1-0-4
Degree $2$
Conductor $704$
Sign $-0.176 - 0.984i$
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  + (0.704 − 0.228i)3-s + (−1.09 − 0.792i)5-s + (−0.503 + 1.54i)7-s + (−1.98 + 1.44i)9-s + (−3.29 + 0.357i)11-s + (2.09 + 2.87i)13-s + (−0.949 − 0.308i)15-s + (−0.0411 + 0.0566i)17-s + (2.45 + 7.56i)19-s + 1.20i·21-s + 5.30i·23-s + (−0.983 − 3.02i)25-s + (−2.37 + 3.26i)27-s + (−3.74 − 1.21i)29-s + (2.17 + 2.98i)31-s + ⋯
L(s)  = 1  + (0.406 − 0.132i)3-s + (−0.487 − 0.354i)5-s + (−0.190 + 0.585i)7-s + (−0.661 + 0.480i)9-s + (−0.994 + 0.107i)11-s + (0.579 + 0.798i)13-s + (−0.245 − 0.0796i)15-s + (−0.00997 + 0.0137i)17-s + (0.563 + 1.73i)19-s + 0.263i·21-s + 1.10i·23-s + (−0.196 − 0.605i)25-s + (−0.456 + 0.628i)27-s + (−0.695 − 0.225i)29-s + (0.389 + 0.536i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.176 - 0.984i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637163 + 0.761756i\)
\(L(\frac12)\) \(\approx\) \(0.637163 + 0.761756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.29 - 0.357i)T \)
good3 \( 1 + (-0.704 + 0.228i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.09 + 0.792i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.503 - 1.54i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.09 - 2.87i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0411 - 0.0566i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.45 - 7.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.30iT - 23T^{2} \)
29 \( 1 + (3.74 + 1.21i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.17 - 2.98i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.12 + 6.54i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-5.50 + 1.78i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.89T + 43T^{2} \)
47 \( 1 + (9.32 - 3.03i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.14 - 2.28i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.62 + 1.50i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.791 - 1.08i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.91iT - 67T^{2} \)
71 \( 1 + (-4.10 + 5.64i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.62 + 3.12i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.85 - 3.52i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.92 - 2.12i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + (5.92 - 4.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.74719249189630248588985549779, −9.670059916735948805868769775622, −8.883679253588936745670944925472, −8.003409227888082667051452891073, −7.58002289470438746738729575143, −6.04213445116854396001883781122, −5.40500703188159770113915274513, −4.12212942561221409649270845560, −3.04112201287339589443026372852, −1.82998857939485835592912563713, 0.47900892112183760300128901823, 2.75184953408054552865797875581, 3.36227234337381777318123160689, 4.59716562489187952382602354908, 5.72555249543062928422036400369, 6.78240300244615109723486915370, 7.69465980750859200970209216568, 8.397821668803353910531794341936, 9.319435375568066596866401057870, 10.24877511501707845978496498499

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