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Properties

Label 2-74-37.10-c1-0-3
Degree $2$
Conductor $74$
Sign $0.150 + 0.988i$
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 + (−1.26 − 2.19i)3-s + (−0.499 − 0.866i)4-s + (−1.97 − 3.42i)5-s + 2.53·6-s + (1.26 + 2.19i)7-s + 0.999·8-s + (−1.71 + 2.96i)9-s + 3.95·10-s + 2.42·11-s + (−1.26 + 2.19i)12-s + (−1 − 1.73i)13-s − 2.53·14-s + (−5.01 + 8.68i)15-s + (−0.5 + 0.866i)16-s + (3.03 − 5.25i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.731 − 1.26i)3-s + (−0.249 − 0.433i)4-s + (−0.884 − 1.53i)5-s + 1.03·6-s + (0.478 + 0.829i)7-s + 0.353·8-s + (−0.570 + 0.987i)9-s + 1.25·10-s + 0.730·11-s + (−0.365 + 0.633i)12-s + (−0.277 − 0.480i)13-s − 0.677·14-s + (−1.29 + 2.24i)15-s + (−0.125 + 0.216i)16-s + (0.735 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.150 + 0.988i$
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: \((2,\ 74,\ (\ :1/2),\ 0.150 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429524 - 0.369205i\)
\(L(\frac12)\) \(\approx\) \(0.429524 - 0.369205i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0222 + 6.08i)T \)
good3 \( 1 + (1.26 + 2.19i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.97 + 3.42i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.26 - 2.19i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.42T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.03 + 5.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.74 - 3.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.955T + 23T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 - 7.48T + 31T^{2} \)
41 \( 1 + (-4.24 - 7.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.11T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (4.21 - 7.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.11 - 3.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.232 + 0.403i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.21 - 5.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.732 - 1.26i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.421T + 73T^{2} \)
79 \( 1 + (1.32 + 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.68 + 2.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.20 + 9.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−14.34934916080836460902366219729, −12.98282146805041547260508613251, −12.03323999143068876605545482314, −11.68882997317237133323940663936, −9.416983798298869430022698571919, −8.225214218751247309642443753991, −7.50971158780724160299084532504, −5.91975186596407756035892670039, −4.87238793835004083883923872346, −1.04701229545292129802348670970, 3.50830827326973275751603700338, 4.44228348321657210616630175821, 6.60238576215399886777773952153, 7.949830522824951245566425204696, 9.770078908630066632342449484864, 10.57438934503834185513821980412, 11.21915110481282342558471383101, 11.97011988276023696699860028343, 14.06007973838360505594506071105, 14.89233620902077356646101164917

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