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Properties

Label 2-74-37.31-c2-0-4
Degree $2$
Conductor $74$
Sign $0.969 + 0.245i$
Analytic cond. $2.01635$
Root an. cond. $1.41998$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s − 3i·3-s + 2i·4-s + (3 − 3i)5-s + (3 − 3i)6-s + 3·7-s + (−2 + 2i)8-s + 6·10-s − 3i·11-s + 6·12-s + (−11 + 11i)13-s + (3 + 3i)14-s + (−9 − 9i)15-s − 4·16-s + (−12 + 12i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − i·3-s + 0.5i·4-s + (0.600 − 0.600i)5-s + (0.5 − 0.5i)6-s + 0.428·7-s + (−0.250 + 0.250i)8-s + 0.600·10-s − 0.272i·11-s + 0.5·12-s + (−0.846 + 0.846i)13-s + (0.214 + 0.214i)14-s + (−0.599 − 0.599i)15-s − 0.250·16-s + (−0.705 + 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(2.01635\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1),\ 0.969 + 0.245i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68860 - 0.210229i\)
\(L(\frac12)\) \(\approx\) \(1.68860 - 0.210229i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
37 \( 1 + (-35 - 12i)T \)
good3 \( 1 + 3iT - 9T^{2} \)
5 \( 1 + (-3 + 3i)T - 25iT^{2} \)
7 \( 1 - 3T + 49T^{2} \)
11 \( 1 + 3iT - 121T^{2} \)
13 \( 1 + (11 - 11i)T - 169iT^{2} \)
17 \( 1 + (12 - 12i)T - 289iT^{2} \)
19 \( 1 + (4 - 4i)T - 361iT^{2} \)
23 \( 1 + (3 - 3i)T - 529iT^{2} \)
29 \( 1 + (-9 - 9i)T + 841iT^{2} \)
31 \( 1 + (32 + 32i)T + 961iT^{2} \)
41 \( 1 - 39iT - 1.68e3T^{2} \)
43 \( 1 + (7 - 7i)T - 1.84e3iT^{2} \)
47 \( 1 - 75T + 2.20e3T^{2} \)
53 \( 1 - 39T + 2.80e3T^{2} \)
59 \( 1 + (-72 + 72i)T - 3.48e3iT^{2} \)
61 \( 1 + (80 + 80i)T + 3.72e3iT^{2} \)
67 \( 1 + 26iT - 4.48e3T^{2} \)
71 \( 1 + 51T + 5.04e3T^{2} \)
73 \( 1 - 25iT - 5.32e3T^{2} \)
79 \( 1 + (28 - 28i)T - 6.24e3iT^{2} \)
83 \( 1 - 27T + 6.88e3T^{2} \)
89 \( 1 + (60 + 60i)T + 7.92e3iT^{2} \)
97 \( 1 + (32 - 32i)T - 9.40e3iT^{2} \)
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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

−14.09615114089837404160116888290, −13.18024768318236724785974196298, −12.52474414266688495155766265550, −11.33828148045788828121916403385, −9.527268961328497167694650829343, −8.235767123782498190808284029472, −7.08906736474636189453862293989, −5.94053369615070102647769069400, −4.48696132384141674100237768971, −1.91439870028612974917874960688, 2.58366342905153102234152014134, 4.31926411675796883593598057922, 5.45850186218706529813164224136, 7.14652667421632709258309298091, 9.119934476401768128778765646387, 10.21644193345398143422413231015, 10.76053856688090007782493526875, 12.13252111689677882783534078303, 13.36704426565477009101049695553, 14.50577344843112693498993970070

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