Properties

Label 2-38-19.11-c9-0-7
Degree $2$
Conductor $38$
Sign $0.974 + 0.224i$
Analytic cond. $19.5713$
Root an. cond. $4.42395$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 13.8i)2-s + (−108. + 188. i)3-s + (−127. − 221. i)4-s + (−480. + 831. i)5-s + (−1.73e3 − 3.01e3i)6-s − 4.30e3·7-s + 4.09e3·8-s + (−1.37e4 − 2.38e4i)9-s + (−7.68e3 − 1.33e4i)10-s − 4.69e4·11-s + 5.56e4·12-s + (−3.51e4 − 6.08e4i)13-s + (3.44e4 − 5.97e4i)14-s + (−1.04e5 − 1.80e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (−1.76e5 + 3.05e5i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.774 + 1.34i)3-s + (−0.249 − 0.433i)4-s + (−0.343 + 0.595i)5-s + (−0.547 − 0.949i)6-s − 0.678·7-s + 0.353·8-s + (−0.701 − 1.21i)9-s + (−0.243 − 0.420i)10-s − 0.967·11-s + 0.774·12-s + (−0.341 − 0.590i)13-s + (0.239 − 0.415i)14-s + (−0.532 − 0.922i)15-s + (−0.125 + 0.216i)16-s + (−0.512 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.974 + 0.224i$
Analytic conductor: \(19.5713\)
Root analytic conductor: \(4.42395\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :9/2),\ 0.974 + 0.224i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.100082 - 0.0114021i\)
\(L(\frac12)\) \(\approx\) \(0.100082 - 0.0114021i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 13.8i)T \)
19 \( 1 + (-5.14e5 - 2.41e5i)T \)
good3 \( 1 + (108. - 188. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (480. - 831. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + 4.30e3T + 4.03e7T^{2} \)
11 \( 1 + 4.69e4T + 2.35e9T^{2} \)
13 \( 1 + (3.51e4 + 6.08e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (1.76e5 - 3.05e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
23 \( 1 + (-4.73e5 - 8.20e5i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 + (1.53e6 + 2.66e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 - 2.95e6T + 2.64e13T^{2} \)
37 \( 1 - 1.34e7T + 1.29e14T^{2} \)
41 \( 1 + (8.44e6 - 1.46e7i)T + (-1.63e14 - 2.83e14i)T^{2} \)
43 \( 1 + (-3.70e4 + 6.41e4i)T + (-2.51e14 - 4.35e14i)T^{2} \)
47 \( 1 + (1.15e7 + 1.99e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (3.19e7 + 5.53e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (4.48e7 - 7.76e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (8.22e7 + 1.42e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-9.56e7 - 1.65e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + (1.80e7 - 3.12e7i)T + (-2.29e16 - 3.97e16i)T^{2} \)
73 \( 1 + (-1.80e8 + 3.13e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-1.48e8 + 2.57e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 6.78e8T + 1.86e17T^{2} \)
89 \( 1 + (3.90e8 + 6.77e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + (-4.56e8 + 7.90e8i)T + (-3.80e17 - 6.58e17i)T^{2} \)
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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

−14.95386109432705215738681165836, −13.13521483170090669386058562297, −11.36024137733651284812266499439, −10.35981798607343621024530697535, −9.577381464323514206653386097499, −7.78698515121273410324459019646, −6.16926444562639548782029810993, −4.98417445290500788235173805978, −3.37192436532995090506676730516, −0.06133525234146195115729137677, 0.903395715356325814153097859911, 2.59027359853039941393736856335, 4.95484654086279728467744288510, 6.70172452555008125871121428999, 7.82607443421162905916090656031, 9.348816938367436335010351575687, 11.00215848966299358302932004523, 12.07686271573679226866381232387, 12.79551136973574367582231483639, 13.70311944415673006118257731704

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