Properties

Label 2-736-184.133-c1-0-5
Degree $2$
Conductor $736$
Sign $0.0715 - 0.997i$
Analytic cond. $5.87698$
Root an. cond. $2.42425$
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  + (1.63 + 1.41i)3-s + (−0.135 − 0.0195i)5-s + (0.888 + 1.94i)7-s + (0.235 + 1.63i)9-s + (1.23 + 4.21i)11-s + (−1.14 − 0.522i)13-s + (−0.193 − 0.223i)15-s + (−0.710 + 0.456i)17-s + (−1.64 + 2.56i)19-s + (−1.30 + 4.42i)21-s + (2.25 − 4.23i)23-s + (−4.77 − 1.40i)25-s + (1.56 − 2.44i)27-s + (3.45 + 5.38i)29-s + (2.47 + 2.85i)31-s + ⋯
L(s)  = 1  + (0.941 + 0.815i)3-s + (−0.0607 − 0.00873i)5-s + (0.335 + 0.735i)7-s + (0.0785 + 0.546i)9-s + (0.373 + 1.27i)11-s + (−0.317 − 0.144i)13-s + (−0.0500 − 0.0577i)15-s + (−0.172 + 0.110i)17-s + (−0.377 + 0.588i)19-s + (−0.283 + 0.966i)21-s + (0.471 − 0.882i)23-s + (−0.955 − 0.280i)25-s + (0.301 − 0.469i)27-s + (0.642 + 0.999i)29-s + (0.444 + 0.512i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0715 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0715 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.0715 - 0.997i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.0715 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50516 + 1.40104i\)
\(L(\frac12)\) \(\approx\) \(1.50516 + 1.40104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-2.25 + 4.23i)T \)
good3 \( 1 + (-1.63 - 1.41i)T + (0.426 + 2.96i)T^{2} \)
5 \( 1 + (0.135 + 0.0195i)T + (4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.888 - 1.94i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-1.23 - 4.21i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.14 + 0.522i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.710 - 0.456i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (1.64 - 2.56i)T + (-7.89 - 17.2i)T^{2} \)
29 \( 1 + (-3.45 - 5.38i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-2.47 - 2.85i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-4.39 + 0.631i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.715 + 4.97i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-5.55 - 4.81i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + (4.41 - 2.01i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-6.76 - 3.08i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-6.47 + 5.60i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (-0.447 + 1.52i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (8.44 + 2.47i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.17 + 1.39i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.635 - 1.39i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (16.4 - 2.36i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (-4.52 + 5.22i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-2.25 + 15.6i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.20889554829039143403805552121, −9.774275755068907833641606282407, −8.824274693596112627225229853112, −8.328533826560310158784829673171, −7.24198280318197971015228519661, −6.15359037645113406989703184860, −4.84409100187075477286650512981, −4.21315433063234219426748148127, −2.99080979143932979526894316058, −1.97777894860014412657548399135, 1.01413494197501676245568627147, 2.37209067621467776195557402167, 3.43845348478528007294025464765, 4.52529996490042889564153227977, 5.90091542133114946681356109069, 6.88974702123591336358962335187, 7.70526270702393530307661644405, 8.296225195253844064918205530451, 9.133451950080294663760807749624, 10.06599435199323384596661759955

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