Properties

Label 2-736-184.101-c1-0-15
Degree $2$
Conductor $736$
Sign $0.840 + 0.542i$
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  + (1.77 − 1.53i)3-s + (3.72 − 0.536i)5-s + (−1.80 + 3.95i)7-s + (0.356 − 2.48i)9-s + (0.791 − 2.69i)11-s + (1.31 − 0.602i)13-s + (5.79 − 6.68i)15-s + (0.426 + 0.274i)17-s + (−1.09 − 1.69i)19-s + (2.87 + 9.79i)21-s + (−4.04 + 2.57i)23-s + (8.82 − 2.59i)25-s + (0.626 + 0.974i)27-s + (−0.637 + 0.991i)29-s + (5.28 − 6.09i)31-s + ⋯
L(s)  = 1  + (1.02 − 0.887i)3-s + (1.66 − 0.239i)5-s + (−0.682 + 1.49i)7-s + (0.118 − 0.826i)9-s + (0.238 − 0.812i)11-s + (0.365 − 0.167i)13-s + (1.49 − 1.72i)15-s + (0.103 + 0.0664i)17-s + (−0.250 − 0.389i)19-s + (0.627 + 2.13i)21-s + (−0.843 + 0.537i)23-s + (1.76 − 0.518i)25-s + (0.120 + 0.187i)27-s + (−0.118 + 0.184i)29-s + (0.948 − 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.840 + 0.542i$
Analytic conductor: \(5.87698\)
Root analytic conductor: \(2.42425\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :1/2),\ 0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52531 - 0.743939i\)
\(L(\frac12)\) \(\approx\) \(2.52531 - 0.743939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.04 - 2.57i)T \)
good3 \( 1 + (-1.77 + 1.53i)T + (0.426 - 2.96i)T^{2} \)
5 \( 1 + (-3.72 + 0.536i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (1.80 - 3.95i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.791 + 2.69i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.31 + 0.602i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.426 - 0.274i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (1.09 + 1.69i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (0.637 - 0.991i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-5.28 + 6.09i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.78 + 0.256i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.06 - 7.38i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-0.318 + 0.275i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + (8.99 + 4.10i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.14 - 0.524i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (6.78 + 5.87i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-0.439 - 1.49i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-3.81 + 1.11i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.83 - 2.46i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-4.36 - 9.55i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-9.43 - 1.35i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (7.96 + 9.18i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.199 - 1.38i)T + (-93.0 + 27.3i)T^{2} \)
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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

−9.821358585383485511823939820251, −9.390455882588825124978695105847, −8.625792615793707920016850387690, −8.019314450772803097635075724133, −6.41497556337756080482148013848, −6.16981819932350577370003048626, −5.19474313161809685578150022523, −3.21073448451061822590306933356, −2.45299393667570360987767813497, −1.58763214473158192855362584977, 1.67840017815724621312247212656, 2.92712679657306610563697300993, 3.89610168083121907722604624548, 4.77838786863468754234716769070, 6.21935754168304329255798897293, 6.81210843511797401217763792955, 8.001798728023578966549090589684, 9.130412407970951455825642102468, 9.670863374425854176578285646182, 10.31661774040259994726058898795

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