Properties

Label 2-736-184.133-c1-0-1
Degree $2$
Conductor $736$
Sign $0.492 - 0.870i$
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  + (−1.38 − 1.20i)3-s + (−2.42 − 0.349i)5-s + (0.504 + 1.10i)7-s + (0.0518 + 0.360i)9-s + (−0.0146 − 0.0499i)11-s + (−1.97 − 0.900i)13-s + (2.94 + 3.40i)15-s + (−1.80 + 1.16i)17-s + (−0.779 + 1.21i)19-s + (0.626 − 2.13i)21-s + (3.84 + 2.86i)23-s + (0.976 + 0.286i)25-s + (−2.61 + 4.06i)27-s + (3.80 + 5.91i)29-s + (1.23 + 1.42i)31-s + ⋯
L(s)  = 1  + (−0.800 − 0.693i)3-s + (−1.08 − 0.156i)5-s + (0.190 + 0.417i)7-s + (0.0172 + 0.120i)9-s + (−0.00442 − 0.0150i)11-s + (−0.546 − 0.249i)13-s + (0.760 + 0.878i)15-s + (−0.438 + 0.281i)17-s + (−0.178 + 0.278i)19-s + (0.136 − 0.465i)21-s + (0.802 + 0.596i)23-s + (0.195 + 0.0573i)25-s + (−0.502 + 0.782i)27-s + (0.705 + 1.09i)29-s + (0.221 + 0.256i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.492 - 0.870i$
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: \((2,\ 736,\ (\ :1/2),\ 0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473699 + 0.276131i\)
\(L(\frac12)\) \(\approx\) \(0.473699 + 0.276131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.84 - 2.86i)T \)
good3 \( 1 + (1.38 + 1.20i)T + (0.426 + 2.96i)T^{2} \)
5 \( 1 + (2.42 + 0.349i)T + (4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.504 - 1.10i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (0.0146 + 0.0499i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.97 + 0.900i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (1.80 - 1.16i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (0.779 - 1.21i)T + (-7.89 - 17.2i)T^{2} \)
29 \( 1 + (-3.80 - 5.91i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-1.23 - 1.42i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-9.88 + 1.42i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (1.47 - 10.2i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-2.92 - 2.53i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 - 2.07T + 47T^{2} \)
53 \( 1 + (5.87 - 2.68i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (-10.5 - 4.81i)T + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (6.59 - 5.71i)T + (8.68 - 60.3i)T^{2} \)
67 \( 1 + (-3.35 + 11.4i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (11.0 + 3.23i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-3.55 - 2.28i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.56 - 3.43i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (13.0 - 1.88i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (9.62 - 11.1i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.06 + 7.39i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.89119300655154730779486080005, −9.648775008586412081126847874316, −8.645030942882352868913930645637, −7.81715612977754231440689430198, −7.05725641315039173407045821589, −6.16944216265054087198138387804, −5.19358746244396760725744053866, −4.20242966302577387430254243422, −2.89146323167313492274623410129, −1.17089175518161471788274161000, 0.36328761054043092338168628814, 2.62004278301184590492955097241, 4.17799125828697352874222695954, 4.49584547158706017679911786608, 5.62723181632620603509568127905, 6.79323181778501964172629451319, 7.58841930828878947539576140164, 8.453691309798819049534040449972, 9.581173490184564464569944850485, 10.40587357973638457673984616836

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