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Properties

Label 2-370-185.103-c1-0-1
Degree $2$
Conductor $370$
Sign $0.0201 - 0.999i$
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  + (−0.5 − 0.866i)2-s + (0.633 + 2.36i)3-s + (−0.499 + 0.866i)4-s + (−1.23 − 1.86i)5-s + (1.73 − 1.73i)6-s + (0.366 + 1.36i)7-s + 0.999·8-s + (−2.59 + 1.50i)9-s + (−1 + 2i)10-s + 6.46i·11-s + (−2.36 − 0.633i)12-s + (0.133 − 0.232i)13-s + (0.999 − i)14-s + (3.63 − 4.09i)15-s + (−0.5 − 0.866i)16-s + (−2.36 + 1.36i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.366 + 1.36i)3-s + (−0.249 + 0.433i)4-s + (−0.550 − 0.834i)5-s + (0.707 − 0.707i)6-s + (0.138 + 0.516i)7-s + 0.353·8-s + (−0.866 + 0.500i)9-s + (−0.316 + 0.632i)10-s + 1.94i·11-s + (−0.683 − 0.183i)12-s + (0.0371 − 0.0643i)13-s + (0.267 − 0.267i)14-s + (0.938 − 1.05i)15-s + (−0.125 − 0.216i)16-s + (−0.573 + 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0201 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0201 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0201 - 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0201 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.700397 + 0.686394i\)
\(L(\frac12)\) \(\approx\) \(0.700397 + 0.686394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good3 \( 1 + (-0.633 - 2.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.366 - 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 6.46iT - 11T^{2} \)
13 \( 1 + (-0.133 + 0.232i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.36 - 1.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.86 + 0.5i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.73T + 23T^{2} \)
29 \( 1 + (5.46 - 5.46i)T - 29iT^{2} \)
31 \( 1 + (-4.19 - 4.19i)T + 31iT^{2} \)
41 \( 1 + (-2.19 - 1.26i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3.09 - 3.09i)T - 47iT^{2} \)
53 \( 1 + (-3.56 + 13.2i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.598 + 2.23i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.5 + 3.09i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.83 - 1.56i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.92 - 7.92i)T - 73iT^{2} \)
79 \( 1 + (-12.9 + 3.46i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.46 + 5.46i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-11.7 - 3.16i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 2.53iT - 97T^{2} \)
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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

−11.54363642196664294772238759877, −10.51573139532303250633496771994, −9.686070683112150101183433308976, −9.142301155507510931916930470188, −8.320236419491236219675546614843, −7.21573427275540525187265570235, −5.21731643457830730884678687689, −4.49062611548685416506908701067, −3.66109746830888744705773493555, −2.04184986262065740661930364118, 0.72997252854848011570873989773, 2.59205667456774069942830768555, 3.95410893442601225765652286778, 5.90966451108915705720335090715, 6.54304275085652480987053295365, 7.60274799173849203082454050892, 7.958002072989437536038516889344, 8.930636467906382105948259880833, 10.31912464273559903138815021417, 11.28244884692883563327661625886

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