Properties

Label 2-38-19.10-c4-0-5
Degree $2$
Conductor $38$
Sign $-0.653 + 0.756i$
Analytic cond. $3.92805$
Root an. cond. $1.98193$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.78 − 0.491i)2-s + (−6.65 − 7.93i)3-s + (7.51 − 2.73i)4-s + (−34.9 − 12.7i)5-s + (−22.4 − 18.8i)6-s + (3.86 − 6.68i)7-s + (19.5 − 11.3i)8-s + (−4.55 + 25.8i)9-s + (−103. − 18.2i)10-s + (−0.839 − 1.45i)11-s + (−71.7 − 41.4i)12-s + (150. − 179. i)13-s + (7.47 − 20.5i)14-s + (131. + 362. i)15-s + (49.0 − 41.1i)16-s + (32.9 + 187. i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.739 − 0.881i)3-s + (0.469 − 0.171i)4-s + (−1.39 − 0.509i)5-s + (−0.623 − 0.522i)6-s + (0.0788 − 0.136i)7-s + (0.306 − 0.176i)8-s + (−0.0561 + 0.318i)9-s + (−1.03 − 0.182i)10-s + (−0.00694 − 0.0120i)11-s + (−0.498 − 0.287i)12-s + (0.892 − 1.06i)13-s + (0.0381 − 0.104i)14-s + (0.586 + 1.61i)15-s + (0.191 − 0.160i)16-s + (0.114 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $-0.653 + 0.756i$
Analytic conductor: \(3.92805\)
Root analytic conductor: \(1.98193\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :2),\ -0.653 + 0.756i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.511043 - 1.11693i\)
\(L(\frac12)\) \(\approx\) \(0.511043 - 1.11693i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.78 + 0.491i)T \)
19 \( 1 + (-242. + 267. i)T \)
good3 \( 1 + (6.65 + 7.93i)T + (-14.0 + 79.7i)T^{2} \)
5 \( 1 + (34.9 + 12.7i)T + (478. + 401. i)T^{2} \)
7 \( 1 + (-3.86 + 6.68i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (0.839 + 1.45i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-150. + 179. i)T + (-4.95e3 - 2.81e4i)T^{2} \)
17 \( 1 + (-32.9 - 187. i)T + (-7.84e4 + 2.85e4i)T^{2} \)
23 \( 1 + (428. - 155. i)T + (2.14e5 - 1.79e5i)T^{2} \)
29 \( 1 + (-1.45e3 - 256. i)T + (6.64e5 + 2.41e5i)T^{2} \)
31 \( 1 + (431. + 249. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.52e3iT - 1.87e6T^{2} \)
41 \( 1 + (1.49e3 + 1.77e3i)T + (-4.90e5 + 2.78e6i)T^{2} \)
43 \( 1 + (1.53e3 + 556. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-483. + 2.74e3i)T + (-4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + (-1.75e3 - 4.81e3i)T + (-6.04e6 + 5.07e6i)T^{2} \)
59 \( 1 + (-662. + 116. i)T + (1.13e7 - 4.14e6i)T^{2} \)
61 \( 1 + (-4.21e3 + 1.53e3i)T + (1.06e7 - 8.89e6i)T^{2} \)
67 \( 1 + (-1.38e3 - 243. i)T + (1.89e7 + 6.89e6i)T^{2} \)
71 \( 1 + (2.90e3 - 7.98e3i)T + (-1.94e7 - 1.63e7i)T^{2} \)
73 \( 1 + (-2.98e3 + 2.50e3i)T + (4.93e6 - 2.79e7i)T^{2} \)
79 \( 1 + (-6.67e3 - 7.95e3i)T + (-6.76e6 + 3.83e7i)T^{2} \)
83 \( 1 + (-1.78e3 + 3.08e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (690. - 822. i)T + (-1.08e7 - 6.17e7i)T^{2} \)
97 \( 1 + (-4.80e3 + 847. i)T + (8.31e7 - 3.02e7i)T^{2} \)
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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

−15.30933648406468988654862124484, −13.59353754895174747297771021209, −12.48460084800552502077302788788, −11.86926472112115847395026906599, −10.76273378942697723973435730889, −8.294405845555632170610546344430, −7.08555252946741810540890024721, −5.54747924043379854637068068845, −3.78869189859731308526852951213, −0.77763622972500541591279648482, 3.64104900771262010218643620056, 4.79651282331831709480444133056, 6.53591438242799240069866807118, 8.111751920474584800783865754385, 10.19520121477112693013343367113, 11.51254735222897372267805244652, 11.84186628011204909581311997362, 13.81868772665099485993277216514, 15.02385267923307460067008932709, 16.10950661721221771292719831100

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