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Properties

Label 2-148-37.10-c3-0-6
Degree $2$
Conductor $148$
Sign $-0.733 + 0.679i$
Analytic cond. $8.73228$
Root an. cond. $2.95504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 − 3.61i)3-s + (5.08 + 8.80i)5-s + (−9.76 − 16.9i)7-s + (4.80 − 8.32i)9-s − 16.2·11-s + (−5.19 − 8.99i)13-s + (21.2 − 36.7i)15-s + (−5.80 + 10.0i)17-s + (−81.4 − 141. i)19-s + (−40.7 + 70.5i)21-s − 132.·23-s + (10.7 − 18.6i)25-s − 152.·27-s − 73.1·29-s − 40.3·31-s + ⋯
L(s)  = 1  + (−0.401 − 0.694i)3-s + (0.454 + 0.787i)5-s + (−0.527 − 0.913i)7-s + (0.178 − 0.308i)9-s − 0.445·11-s + (−0.110 − 0.191i)13-s + (0.364 − 0.632i)15-s + (−0.0828 + 0.143i)17-s + (−0.983 − 1.70i)19-s + (−0.423 + 0.732i)21-s − 1.20·23-s + (0.0862 − 0.149i)25-s − 1.08·27-s − 0.468·29-s − 0.233·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(8.73228\)
Root analytic conductor: \(2.95504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :3/2),\ -0.733 + 0.679i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.332603 - 0.847811i\)
\(L(\frac12)\) \(\approx\) \(0.332603 - 0.847811i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (17.3 + 224. i)T \)
good3 \( 1 + (2.08 + 3.61i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.08 - 8.80i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (9.76 + 16.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 16.2T + 1.33e3T^{2} \)
13 \( 1 + (5.19 + 8.99i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (5.80 - 10.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (81.4 + 141. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 132.T + 1.21e4T^{2} \)
29 \( 1 + 73.1T + 2.43e4T^{2} \)
31 \( 1 + 40.3T + 2.97e4T^{2} \)
41 \( 1 + (-76.9 - 133. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 367.T + 7.95e4T^{2} \)
47 \( 1 - 529.T + 1.03e5T^{2} \)
53 \( 1 + (104. - 181. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (78.9 - 136. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-93.5 - 162. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (151. + 261. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-143. - 247. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 4.69T + 3.89e5T^{2} \)
79 \( 1 + (-402. - 697. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-623. + 1.08e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-110. + 191. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 91.3T + 9.12e5T^{2} \)
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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

−12.38119133698774248032580402947, −11.01712603942720625355575945403, −10.36875410456401446092999962242, −9.220034748984804418055402737014, −7.54664549607335445469727329542, −6.79656648322873928180545608997, −5.93905428414852956932013107091, −4.09008314921524380363242271878, −2.42967514705288636615344046622, −0.44216712899625466789945380963, 2.06881618693417552496568981042, 4.05433968668879542524220711799, 5.32900541909227601895282527692, 6.05653972982032825969262004298, 7.88161605941273390808704963821, 9.034919810032489439473036641722, 9.887745025046572546920770905627, 10.75456508868069383429708985210, 12.14256733444255783922312124060, 12.75107186660359337945807902658

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