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Properties

Label 2-74-37.31-c6-0-2
Degree $2$
Conductor $74$
Sign $-0.388 + 0.921i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 4i)2-s + 12.3i·3-s + 32i·4-s + (−142. + 142. i)5-s + (−49.4 + 49.4i)6-s − 327.·7-s + (−128 + 128i)8-s + 576.·9-s − 1.13e3·10-s − 1.87e3i·11-s − 395.·12-s + (471. − 471. i)13-s + (−1.30e3 − 1.30e3i)14-s + (−1.75e3 − 1.75e3i)15-s − 1.02e3·16-s + (1.65e3 − 1.65e3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.457i·3-s + 0.5i·4-s + (−1.13 + 1.13i)5-s + (−0.228 + 0.228i)6-s − 0.954·7-s + (−0.250 + 0.250i)8-s + 0.790·9-s − 1.13·10-s − 1.40i·11-s − 0.228·12-s + (0.214 − 0.214i)13-s + (−0.477 − 0.477i)14-s + (−0.520 − 0.520i)15-s − 0.250·16-s + (0.336 − 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ -0.388 + 0.921i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.194187 - 0.292512i\)
\(L(\frac12)\) \(\approx\) \(0.194187 - 0.292512i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
37 \( 1 + (2.34e4 - 4.48e4i)T \)
good3 \( 1 - 12.3iT - 729T^{2} \)
5 \( 1 + (142. - 142. i)T - 1.56e4iT^{2} \)
7 \( 1 + 327.T + 1.17e5T^{2} \)
11 \( 1 + 1.87e3iT - 1.77e6T^{2} \)
13 \( 1 + (-471. + 471. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-1.65e3 + 1.65e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (331. - 331. i)T - 4.70e7iT^{2} \)
23 \( 1 + (1.65e4 - 1.65e4i)T - 1.48e8iT^{2} \)
29 \( 1 + (4.64e3 + 4.64e3i)T + 5.94e8iT^{2} \)
31 \( 1 + (2.83e4 + 2.83e4i)T + 8.87e8iT^{2} \)
41 \( 1 + 7.09e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.85e4 - 2.85e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 4.76e4T + 1.07e10T^{2} \)
53 \( 1 - 1.67e4T + 2.21e10T^{2} \)
59 \( 1 + (1.60e5 - 1.60e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.23e5 + 1.23e5i)T + 5.15e10iT^{2} \)
67 \( 1 - 1.32e4iT - 9.04e10T^{2} \)
71 \( 1 - 3.32e5T + 1.28e11T^{2} \)
73 \( 1 - 6.56e5iT - 1.51e11T^{2} \)
79 \( 1 + (5.70e5 - 5.70e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 7.17e5T + 3.26e11T^{2} \)
89 \( 1 + (1.76e5 + 1.76e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (-1.15e6 + 1.15e6i)T - 8.32e11iT^{2} \)
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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

−14.20691346883348368208751226927, −13.20779117696895508686421454058, −11.86205687604654831278256254500, −10.91299561159258060400561826169, −9.698768272042949027898626558435, −8.031298080141289892418670709683, −7.02509284207138451303547739541, −5.84168630413271452356125719376, −3.85153118322949861252021096800, −3.27985972750267149448050929051, 0.11294397825552101396264758071, 1.69403734275696581328741553371, 3.80855792088860023222785819819, 4.74414250899483247076335987750, 6.60842470466384977472519416305, 7.82579442487273718606101826305, 9.273037313975595394563628195039, 10.40432596132716976002414157840, 12.18286232985742332159527105668, 12.40479241315099643467011567356

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