Properties

Label 2-38-19.6-c7-0-10
Degree $2$
Conductor $38$
Sign $-0.238 - 0.971i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.12 − 5.14i)2-s + (−67.0 − 24.3i)3-s + (11.1 − 63.0i)4-s + (−63.2 − 358. i)5-s + (−536. + 195. i)6-s + (−158. + 274. i)7-s + (−256. − 443. i)8-s + (2.22e3 + 1.86e3i)9-s + (−2.23e3 − 1.87e3i)10-s + (1.52e3 + 2.63e3i)11-s + (−2.28e3 + 3.95e3i)12-s + (−3.92e3 + 1.42e3i)13-s + (440. + 2.49e3i)14-s + (−4.50e3 + 2.55e4i)15-s + (−3.84e3 − 1.40e3i)16-s + (−2.29e4 + 1.92e4i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−1.43 − 0.521i)3-s + (0.0868 − 0.492i)4-s + (−0.226 − 1.28i)5-s + (−1.01 + 0.368i)6-s + (−0.174 + 0.302i)7-s + (−0.176 − 0.306i)8-s + (1.01 + 0.852i)9-s + (−0.705 − 0.592i)10-s + (0.345 + 0.597i)11-s + (−0.381 + 0.660i)12-s + (−0.494 + 0.180i)13-s + (0.0428 + 0.243i)14-s + (−0.345 + 1.95i)15-s + (−0.234 − 0.0855i)16-s + (−1.13 + 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.238 - 0.971i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ -0.238 - 0.971i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.119297 + 0.152127i\)
\(L(\frac12)\) \(\approx\) \(0.119297 + 0.152127i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-6.12 + 5.14i)T \)
19 \( 1 + (-2.97e4 + 2.42e3i)T \)
good3 \( 1 + (67.0 + 24.3i)T + (1.67e3 + 1.40e3i)T^{2} \)
5 \( 1 + (63.2 + 358. i)T + (-7.34e4 + 2.67e4i)T^{2} \)
7 \( 1 + (158. - 274. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-1.52e3 - 2.63e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (3.92e3 - 1.42e3i)T + (4.80e7 - 4.03e7i)T^{2} \)
17 \( 1 + (2.29e4 - 1.92e4i)T + (7.12e7 - 4.04e8i)T^{2} \)
23 \( 1 + (-1.01e4 + 5.75e4i)T + (-3.19e9 - 1.16e9i)T^{2} \)
29 \( 1 + (7.88e4 + 6.61e4i)T + (2.99e9 + 1.69e10i)T^{2} \)
31 \( 1 + (1.04e5 - 1.81e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 1.57e4T + 9.49e10T^{2} \)
41 \( 1 + (7.23e5 + 2.63e5i)T + (1.49e11 + 1.25e11i)T^{2} \)
43 \( 1 + (1.33e5 + 7.57e5i)T + (-2.55e11 + 9.29e10i)T^{2} \)
47 \( 1 + (4.37e5 + 3.67e5i)T + (8.79e10 + 4.98e11i)T^{2} \)
53 \( 1 + (2.01e5 - 1.14e6i)T + (-1.10e12 - 4.01e11i)T^{2} \)
59 \( 1 + (4.79e5 - 4.02e5i)T + (4.32e11 - 2.45e12i)T^{2} \)
61 \( 1 + (5.50e5 - 3.12e6i)T + (-2.95e12 - 1.07e12i)T^{2} \)
67 \( 1 + (1.01e6 + 8.48e5i)T + (1.05e12 + 5.96e12i)T^{2} \)
71 \( 1 + (4.94e5 + 2.80e6i)T + (-8.54e12 + 3.11e12i)T^{2} \)
73 \( 1 + (4.01e6 + 1.46e6i)T + (8.46e12 + 7.10e12i)T^{2} \)
79 \( 1 + (-5.63e6 - 2.05e6i)T + (1.47e13 + 1.23e13i)T^{2} \)
83 \( 1 + (2.92e6 - 5.07e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (1.16e6 - 4.23e5i)T + (3.38e13 - 2.84e13i)T^{2} \)
97 \( 1 + (1.81e6 - 1.52e6i)T + (1.40e13 - 7.95e13i)T^{2} \)
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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

−13.38099569732020707036152563614, −12.33192167811761596658021323218, −12.01219320145092313666975231967, −10.61013119341797807885147629275, −8.965048480835245295092606625302, −6.89139499506533559749107147717, −5.50002852897610917863447328387, −4.48462210898595668282420511186, −1.57078154209341050579205143055, −0.085736822073696322175387795826, 3.37519194203176235225902307776, 5.02511236711084448795262216438, 6.33990331513798295837353103968, 7.30353385810909288275562432850, 9.763486293523787915436708612011, 11.21051049018709451630821562540, 11.55219530214636272629317090228, 13.33842028820500533049997656683, 14.63970534227204213528622666631, 15.70562431752044588799888627594

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