Properties

Label 2-74-37.10-c1-0-1
Degree $2$
Conductor $74$
Sign $0.0800 - 0.996i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.52 + 2.64i)3-s + (−0.499 − 0.866i)4-s + (−0.629 − 1.09i)5-s − 3.05·6-s + (−1.52 − 2.64i)7-s + 0.999·8-s + (−3.15 + 5.46i)9-s + 1.25·10-s + 5.31·11-s + (1.52 − 2.64i)12-s + (−1 − 1.73i)13-s + 3.05·14-s + (1.92 − 3.32i)15-s + (−0.5 + 0.866i)16-s + (−2.55 + 4.41i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.880 + 1.52i)3-s + (−0.249 − 0.433i)4-s + (−0.281 − 0.487i)5-s − 1.24·6-s + (−0.576 − 0.998i)7-s + 0.353·8-s + (−1.05 + 1.82i)9-s + 0.398·10-s + 1.60·11-s + (0.440 − 0.762i)12-s + (−0.277 − 0.480i)13-s + 0.815·14-s + (0.496 − 0.859i)15-s + (−0.125 + 0.216i)16-s + (−0.618 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.0800 - 0.996i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.0800 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679653 + 0.627273i\)
\(L(\frac12)\) \(\approx\) \(0.679653 + 0.627273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (1.37 - 5.92i)T \)
good3 \( 1 + (-1.52 - 2.64i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.629 + 1.09i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.52 + 2.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.31T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.55 - 4.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.39 + 4.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 + 4.05T + 29T^{2} \)
31 \( 1 + 0.791T + 31T^{2} \)
41 \( 1 + (-0.104 - 0.180i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.36T + 43T^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + (5.65 - 9.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.36 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.02 + 5.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.65 - 8.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.52 - 6.10i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.31T + 73T^{2} \)
79 \( 1 + (-5.70 - 9.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.78 + 3.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.63 - 2.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.20T + 97T^{2} \)
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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

−15.00987700516465633440538993369, −14.19773089507496903442810739688, −13.03222204767425692862954533230, −11.05802304320704398615953571512, −10.03470392558039367002711832799, −9.138490409215562991005082762445, −8.317900127744862278149754496246, −6.67466518249955352430153828515, −4.64489223391926068959373998079, −3.72835324663060223169942872204, 2.01186820963435793697241308306, 3.40537620758945713518790439201, 6.43184687677280413923488605373, 7.32487999167103792420946246589, 8.786493277731485260742185958817, 9.354247945384170293962122223822, 11.47030751830343735009271198998, 12.15275900688051499230355389307, 13.04654929620210168742045784337, 14.20592807727266602175310584177

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