Properties

Label 2-370-185.84-c1-0-9
Degree $2$
Conductor $370$
Sign $0.388 - 0.921i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.133 + 2.23i)5-s + (3.46 − 2i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1 + 1.99i)10-s − 3·11-s + (0.866 − 0.5i)13-s + 3.99·14-s + (−0.5 + 0.866i)16-s + (5.19 + 3i)17-s + (−2.59 + 1.5i)18-s + (−1.5 − 2.59i)19-s + (−1.86 + 1.23i)20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.0599 + 0.998i)5-s + (1.30 − 0.755i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.316 + 0.632i)10-s − 0.904·11-s + (0.240 − 0.138i)13-s + 1.06·14-s + (−0.125 + 0.216i)16-s + (1.26 + 0.727i)17-s + (−0.612 + 0.353i)18-s + (−0.344 − 0.596i)19-s + (−0.417 + 0.275i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67349 + 1.11077i\)
\(L(\frac12)\) \(\approx\) \(1.67349 + 1.11077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.133 - 2.23i)T \)
37 \( 1 + (2.59 + 5.5i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.46 + 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 + (8.66 + 5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.19 + 3i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−11.25710906759082153748893159995, −10.91118193526816205884873664093, −10.11839648699964603588269951931, −8.179697184492945270803829271806, −7.903714484017101141703843967704, −6.85941695458242684411104117143, −5.60962180821293832263610493412, −4.79364145498103143262749446962, −3.46208207938565518537077697797, −2.15608283953491981811341272537, 1.34375886735816440452922574959, 2.85172604290185480837619227158, 4.35823241665747837807536057036, 5.33810577372484577096069110447, 5.88272133762083988293051309975, 7.64574026710054425710151173162, 8.497420863063304553471086642147, 9.327238437367939281671064719475, 10.46873013268013731120859565938, 11.58141739057250108519038847514

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