Properties

Label 2-74-37.33-c5-0-2
Degree $2$
Conductor $74$
Sign $-0.910 - 0.413i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.694 − 3.93i)2-s + (3.82 + 21.6i)3-s + (−15.0 − 5.47i)4-s + (−48.1 + 40.4i)5-s + 88.0·6-s + (134. − 112. i)7-s + (−32 + 55.4i)8-s + (−226. + 82.4i)9-s + (125. + 217. i)10-s + (−241. + 418. i)11-s + (61.1 − 346. i)12-s + (−625. − 227. i)13-s + (−350. − 607. i)14-s + (−1.06e3 − 889. i)15-s + (196. + 164. i)16-s + (−2.16e3 + 789. i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.245 + 1.38i)3-s + (−0.469 − 0.171i)4-s + (−0.861 + 0.723i)5-s + 0.998·6-s + (1.03 − 0.869i)7-s + (−0.176 + 0.306i)8-s + (−0.932 + 0.339i)9-s + (0.397 + 0.689i)10-s + (−0.601 + 1.04i)11-s + (0.122 − 0.694i)12-s + (−1.02 − 0.373i)13-s + (−0.478 − 0.827i)14-s + (−1.21 − 1.02i)15-s + (0.191 + 0.160i)16-s + (−1.81 + 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.910 - 0.413i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.910 - 0.413i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.146575 + 0.677684i\)
\(L(\frac12)\) \(\approx\) \(0.146575 + 0.677684i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.694 + 3.93i)T \)
37 \( 1 + (-5.08e3 - 6.59e3i)T \)
good3 \( 1 + (-3.82 - 21.6i)T + (-228. + 83.1i)T^{2} \)
5 \( 1 + (48.1 - 40.4i)T + (542. - 3.07e3i)T^{2} \)
7 \( 1 + (-134. + 112. i)T + (2.91e3 - 1.65e4i)T^{2} \)
11 \( 1 + (241. - 418. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (625. + 227. i)T + (2.84e5 + 2.38e5i)T^{2} \)
17 \( 1 + (2.16e3 - 789. i)T + (1.08e6 - 9.12e5i)T^{2} \)
19 \( 1 + (145. + 826. i)T + (-2.32e6 + 8.46e5i)T^{2} \)
23 \( 1 + (2.15e3 + 3.73e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.18e3 - 3.79e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 5.34e3T + 2.86e7T^{2} \)
41 \( 1 + (-1.96e3 - 716. i)T + (8.87e7 + 7.44e7i)T^{2} \)
43 \( 1 - 8.76e3T + 1.47e8T^{2} \)
47 \( 1 + (-3.51e3 - 6.08e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.70e3 + 3.94e3i)T + (7.26e7 + 4.11e8i)T^{2} \)
59 \( 1 + (4.49e3 + 3.76e3i)T + (1.24e8 + 7.04e8i)T^{2} \)
61 \( 1 + (-3.26e4 - 1.18e4i)T + (6.46e8 + 5.42e8i)T^{2} \)
67 \( 1 + (5.21e4 - 4.37e4i)T + (2.34e8 - 1.32e9i)T^{2} \)
71 \( 1 + (-1.15e4 - 6.55e4i)T + (-1.69e9 + 6.17e8i)T^{2} \)
73 \( 1 + 7.14e4T + 2.07e9T^{2} \)
79 \( 1 + (2.03e4 - 1.70e4i)T + (5.34e8 - 3.03e9i)T^{2} \)
83 \( 1 + (-1.44e4 + 5.24e3i)T + (3.01e9 - 2.53e9i)T^{2} \)
89 \( 1 + (1.35e4 + 1.14e4i)T + (9.69e8 + 5.49e9i)T^{2} \)
97 \( 1 + (-1.32e4 - 2.29e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−14.45034626054345818166215008534, −12.87911848254529117808657755019, −11.42391558431021712726198094684, −10.64553765156254568437337130503, −10.06107042097731649767678230932, −8.517620013160368305731053864787, −7.24447298113649502266708982014, −4.58060316642954519037659125051, −4.31478097158776875003061764432, −2.56818449480577933393242694371, 0.27033145027933941987130918008, 2.19211344111157872249899527138, 4.58724213784751775439622351686, 5.94392723842167412179892773350, 7.52156927793336295413554135502, 8.092977101514814403004971715811, 9.013431781311544684570665036146, 11.50377973216242104282661312968, 12.09826563948127769367815213622, 13.25103069980080601262527796587

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