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Properties

Label 2-444-37.36-c1-0-2
Degree $2$
Conductor $444$
Sign $0.367 - 0.929i$
Analytic cond. $3.54535$
Root an. cond. $1.88291$
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  − 3-s + 0.540i·5-s − 1.23·7-s + 9-s + 2.47·11-s + 4.57i·13-s − 0.540i·15-s + 3.36i·17-s − 1.74i·19-s + 1.23·21-s + 8.61i·23-s + 4.70·25-s − 27-s + 7.94i·29-s − 6.32i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.241i·5-s − 0.467·7-s + 0.333·9-s + 0.745·11-s + 1.26i·13-s − 0.139i·15-s + 0.817i·17-s − 0.401i·19-s + 0.269·21-s + 1.79i·23-s + 0.941·25-s − 0.192·27-s + 1.47i·29-s − 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851188 + 0.578813i\)
\(L(\frac12)\) \(\approx\) \(0.851188 + 0.578813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
37 \( 1 + (2.23 - 5.65i)T \)
good5 \( 1 - 0.540iT - 5T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 - 4.57iT - 13T^{2} \)
17 \( 1 - 3.36iT - 17T^{2} \)
19 \( 1 + 1.74iT - 19T^{2} \)
23 \( 1 - 8.61iT - 23T^{2} \)
29 \( 1 - 7.94iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 2.82iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 2.28iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 + 7.70T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 3.23T + 73T^{2} \)
79 \( 1 + 9.56iT - 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + 8.61iT - 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−11.37966062427594909450055997658, −10.46509992291262415605389160912, −9.473598215398564775073956892605, −8.787792774062817118904772995279, −7.29850769993290821048598606178, −6.65258134323036059923835904525, −5.72459682183535992472364605042, −4.46177293159658183100883153492, −3.39839459191073237504345175532, −1.59865815106246346437730187120, 0.74837225273034262343388348330, 2.76417633886699127830832873462, 4.14919994336058509881377221853, 5.24761690332822063716149584052, 6.22526793826310887430262328071, 7.08412817060963723399664858011, 8.250044339699003410553665193349, 9.189857645882714803737006967127, 10.19385244077953815606123880899, 10.82523269546505448556699746842

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