Properties

Label 2-74-37.31-c4-0-10
Degree $2$
Conductor $74$
Sign $-0.606 + 0.795i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
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 − 2i)2-s − 10.5i·3-s + 8i·4-s + (30.2 − 30.2i)5-s + (−21.0 + 21.0i)6-s + 69.0·7-s + (16 − 16i)8-s − 29.6·9-s − 120.·10-s + 10.9i·11-s + 84.1·12-s + (80.0 − 80.0i)13-s + (−138. − 138. i)14-s + (−318. − 318. i)15-s − 64·16-s + (−345. + 345. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 1.16i·3-s + 0.5i·4-s + (1.20 − 1.20i)5-s + (−0.584 + 0.584i)6-s + 1.40·7-s + (0.250 − 0.250i)8-s − 0.365·9-s − 1.20·10-s + 0.0908i·11-s + 0.584·12-s + (0.473 − 0.473i)13-s + (−0.704 − 0.704i)14-s + (−1.41 − 1.41i)15-s − 0.250·16-s + (−1.19 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.606 + 0.795i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.774812 - 1.56528i\)
\(L(\frac12)\) \(\approx\) \(0.774812 - 1.56528i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
37 \( 1 + (917. - 1.01e3i)T \)
good3 \( 1 + 10.5iT - 81T^{2} \)
5 \( 1 + (-30.2 + 30.2i)T - 625iT^{2} \)
7 \( 1 - 69.0T + 2.40e3T^{2} \)
11 \( 1 - 10.9iT - 1.46e4T^{2} \)
13 \( 1 + (-80.0 + 80.0i)T - 2.85e4iT^{2} \)
17 \( 1 + (345. - 345. i)T - 8.35e4iT^{2} \)
19 \( 1 + (-121. + 121. i)T - 1.30e5iT^{2} \)
23 \( 1 + (590. - 590. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-49.9 - 49.9i)T + 7.07e5iT^{2} \)
31 \( 1 + (-947. - 947. i)T + 9.23e5iT^{2} \)
41 \( 1 - 3.12e3iT - 2.82e6T^{2} \)
43 \( 1 + (-522. + 522. i)T - 3.41e6iT^{2} \)
47 \( 1 - 315.T + 4.87e6T^{2} \)
53 \( 1 + 3.94e3T + 7.89e6T^{2} \)
59 \( 1 + (-3.55e3 + 3.55e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (519. + 519. i)T + 1.38e7iT^{2} \)
67 \( 1 + 3.73e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.19e3T + 2.54e7T^{2} \)
73 \( 1 + 1.21e3iT - 2.83e7T^{2} \)
79 \( 1 + (1.83e3 - 1.83e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 1.53e3T + 4.74e7T^{2} \)
89 \( 1 + (5.11e3 + 5.11e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (4.01e3 - 4.01e3i)T - 8.85e7iT^{2} \)
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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

−13.27483033667788077492938058178, −12.46749536524070560896133116661, −11.37077397424118757518563024589, −10.00554483299274852635592476427, −8.617316953628916391394798707416, −7.990308326956519671556349567688, −6.26050823515227891069350370986, −4.78432131872727004039885877458, −1.86728897107088372558998871780, −1.28547910699209713373851189589, 2.17613300678973335575849843622, 4.44927885833823026004442366468, 5.79137329609808236942747385914, 7.11188818069892158041418494378, 8.722503602105646054259797566138, 9.795213792714066640640965797795, 10.63500572597066570318911233270, 11.38366178946991636486286455645, 13.87976175284747651376862208975, 14.27133553058726278660950425593

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