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Properties

Label 2-738-41.5-c1-0-1
Degree $2$
Conductor $738$
Sign $-0.448 - 0.893i$
Analytic cond. $5.89295$
Root an. cond. $2.42754$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−1.42 − 0.461i)5-s + (0.476 + 3.01i)7-s + (0.951 − 0.309i)8-s + (0.461 + 1.42i)10-s + (0.776 − 1.52i)11-s + (−0.571 − 0.0904i)13-s + (2.15 − 2.15i)14-s + (−0.809 − 0.587i)16-s + (−2.36 − 1.20i)17-s + (−4.69 + 0.744i)19-s + (0.878 − 1.20i)20-s + (−1.68 + 0.267i)22-s + (−1.24 + 0.907i)23-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.635 − 0.206i)5-s + (0.180 + 1.13i)7-s + (0.336 − 0.109i)8-s + (0.146 + 0.449i)10-s + (0.234 − 0.459i)11-s + (−0.158 − 0.0250i)13-s + (0.576 − 0.576i)14-s + (−0.202 − 0.146i)16-s + (−0.573 − 0.292i)17-s + (−1.07 + 0.170i)19-s + (0.196 − 0.270i)20-s + (−0.360 + 0.0570i)22-s + (−0.260 + 0.189i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.448 - 0.893i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.187238 + 0.303427i\)
\(L(\frac12)\) \(\approx\) \(0.187238 + 0.303427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 \)
41 \( 1 + (-5.94 + 2.36i)T \)
good5 \( 1 + (1.42 + 0.461i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.476 - 3.01i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (-0.776 + 1.52i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (0.571 + 0.0904i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.36 + 1.20i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (4.69 - 0.744i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (1.24 - 0.907i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (8.90 - 4.53i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-2.70 - 8.32i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.69 - 8.28i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (2.38 + 3.28i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.216 - 1.36i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (8.40 - 4.28i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.877 - 0.637i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.609 + 0.839i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.40 - 6.67i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-0.679 + 1.33i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 + 2.88iT - 73T^{2} \)
79 \( 1 + (4.24 + 4.24i)T + 79iT^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (-0.893 - 5.64i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (3.64 + 7.14i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.80337675856323766106445852231, −9.742258887998639606588252333827, −8.744778727263001829969168353840, −8.494668745800872988515037049792, −7.38601343418313017049400745596, −6.26285243078003574514105958593, −5.15428232375594816948146297377, −4.08724465008524405800158683809, −2.96434307774776784497637123208, −1.76777781301945120287539751628, 0.20319206154699238433557513856, 2.01964546294478859199111792179, 3.93643099938520062975410409023, 4.39177916302840595396960576986, 5.85283505513251381434412785761, 6.80524575737184421137487335665, 7.56156187742519873528795106833, 8.101157716660916692722777022547, 9.268997952289338078733397886008, 9.987207848993183103676813501389

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