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Properties

Label 2-74-37.31-c2-0-2
Degree $2$
Conductor $74$
Sign $-0.0824 - 0.996i$
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 + 4i·3-s + 2i·4-s + (3 − 3i)5-s + (−4 + 4i)6-s − 4·7-s + (−2 + 2i)8-s − 7·9-s + 6·10-s + 4i·11-s − 8·12-s + (3 − 3i)13-s + (−4 − 4i)14-s + (12 + 12i)15-s − 4·16-s + (23 − 23i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 1.33i·3-s + 0.5i·4-s + (0.600 − 0.600i)5-s + (−0.666 + 0.666i)6-s − 0.571·7-s + (−0.250 + 0.250i)8-s − 0.777·9-s + 0.600·10-s + 0.363i·11-s − 0.666·12-s + (0.230 − 0.230i)13-s + (−0.285 − 0.285i)14-s + (0.800 + 0.800i)15-s − 0.250·16-s + (1.35 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0824 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0824 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.0824 - 0.996i$
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: \((2,\ 74,\ (\ :1),\ -0.0824 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10398 + 1.19913i\)
\(L(\frac12)\) \(\approx\) \(1.10398 + 1.19913i\)
\(L(2)\) 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 \( 1 + 37iT \)
good3 \( 1 - 4iT - 9T^{2} \)
5 \( 1 + (-3 + 3i)T - 25iT^{2} \)
7 \( 1 + 4T + 49T^{2} \)
11 \( 1 - 4iT - 121T^{2} \)
13 \( 1 + (-3 + 3i)T - 169iT^{2} \)
17 \( 1 + (-23 + 23i)T - 289iT^{2} \)
19 \( 1 + (-10 + 10i)T - 361iT^{2} \)
23 \( 1 + (10 - 10i)T - 529iT^{2} \)
29 \( 1 + (19 + 19i)T + 841iT^{2} \)
31 \( 1 + (18 + 18i)T + 961iT^{2} \)
41 \( 1 - 74iT - 1.68e3T^{2} \)
43 \( 1 + (-42 + 42i)T - 1.84e3iT^{2} \)
47 \( 1 + 44T + 2.20e3T^{2} \)
53 \( 1 + 80T + 2.80e3T^{2} \)
59 \( 1 + (54 - 54i)T - 3.48e3iT^{2} \)
61 \( 1 + (3 + 3i)T + 3.72e3iT^{2} \)
67 \( 1 + 12iT - 4.48e3T^{2} \)
71 \( 1 - 124T + 5.04e3T^{2} \)
73 \( 1 + 10iT - 5.32e3T^{2} \)
79 \( 1 + (-14 + 14i)T - 6.24e3iT^{2} \)
83 \( 1 + 64T + 6.88e3T^{2} \)
89 \( 1 + (-17 - 17i)T + 7.92e3iT^{2} \)
97 \( 1 + (-129 + 129i)T - 9.40e3iT^{2} \)
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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

−14.74582601661900919183475078088, −13.67807641372599725004753761468, −12.65572488415663634665699921647, −11.29772222717272103930219538802, −9.680052062216690319669824061045, −9.401469888143138558411485371251, −7.56444323665446958086294095506, −5.76186677957676864048926210040, −4.84866898617643768664888855560, −3.37875699457624295053014397407, 1.65956295605923317123107545788, 3.33921956009323824632251740169, 5.83275302681272204421359771488, 6.61779079946939470987437219179, 8.032794725847557928015040339350, 9.776008111707046971156775444856, 10.84711011935171845950599526817, 12.28436777667639013895495109900, 12.80080056819426928365159945434, 13.94971302798630805091721976874

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