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Properties

Label 2-704-11.3-c1-0-17
Degree $2$
Conductor $704$
Sign $-0.714 + 0.699i$
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  + (−0.476 − 1.46i)3-s + (−0.309 − 0.224i)5-s + (−0.476 + 1.46i)7-s + (0.500 − 0.363i)9-s + (−1.54 − 2.93i)11-s + (1.30 − 0.951i)13-s + (−0.182 + 0.560i)15-s + (0.690 + 0.502i)17-s + (−1.43 − 4.40i)19-s + 2.38·21-s − 4.99·23-s + (−1.5 − 4.61i)25-s + (−4.51 − 3.28i)27-s + (0.0450 − 0.138i)29-s + (1.72 − 1.25i)31-s + ⋯
L(s)  = 1  + (−0.275 − 0.847i)3-s + (−0.138 − 0.100i)5-s + (−0.180 + 0.554i)7-s + (0.166 − 0.121i)9-s + (−0.465 − 0.885i)11-s + (0.363 − 0.263i)13-s + (−0.0470 + 0.144i)15-s + (0.167 + 0.121i)17-s + (−0.328 − 1.01i)19-s + 0.519·21-s − 1.04·23-s + (−0.300 − 0.923i)25-s + (−0.869 − 0.631i)27-s + (0.00837 − 0.0257i)29-s + (0.309 − 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.714 + 0.699i$
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: \((2,\ 704,\ (\ :1/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364068 - 0.892788i\)
\(L(\frac12)\) \(\approx\) \(0.364068 - 0.892788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 + 2.93i)T \)
good3 \( 1 + (0.476 + 1.46i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.309 + 0.224i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.476 - 1.46i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.30 + 0.951i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.690 - 0.502i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.43 + 4.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.99T + 23T^{2} \)
29 \( 1 + (-0.0450 + 0.138i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.72 + 1.25i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.572 - 1.76i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.190 - 0.587i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.90T + 43T^{2} \)
47 \( 1 + (2.97 + 9.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.92 - 5.03i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.19 - 9.84i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.1 + 7.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.08T + 67T^{2} \)
71 \( 1 + (8.85 + 6.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.04 + 9.37i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-12.8 + 9.36i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.85 - 6.43i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.18T + 89T^{2} \)
97 \( 1 + (-2.54 + 1.84i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19886021252197563334047008689, −9.136863945117235631369064546811, −8.265951946335125427008440181377, −7.57133357731916593752976635300, −6.35565293255664258745317735789, −5.97515262635844529088067980730, −4.68087505874833688286013252063, −3.35586778178154111868546731879, −2.10093568625273205453620711155, −0.51475004759913468993907281840, 1.82563791655810175275408017663, 3.53201826022431532899852490318, 4.29355748824312498721325615222, 5.19100857658609409039012642993, 6.27776930822922132147581642079, 7.37606108957641052798412403711, 8.051360144298623996041990641783, 9.371191403847613401766427830942, 10.01556849682875451946125733818, 10.57552582490976619794797208692

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