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Properties

Label 2-370-185.103-c1-0-11
Degree $2$
Conductor $370$
Sign $-0.146 + 0.989i$
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 + (−0.176 − 0.658i)3-s + (−0.499 + 0.866i)4-s + (0.242 − 2.22i)5-s + (−0.482 + 0.482i)6-s + (1.09 + 4.10i)7-s + 0.999·8-s + (2.19 − 1.26i)9-s + (−2.04 + 0.901i)10-s − 2.14i·11-s + (0.658 + 0.176i)12-s + (2.57 − 4.46i)13-s + (3.00 − 3.00i)14-s + (−1.50 + 0.232i)15-s + (−0.5 − 0.866i)16-s + (2.78 − 1.60i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.101 − 0.380i)3-s + (−0.249 + 0.433i)4-s + (0.108 − 0.994i)5-s + (−0.196 + 0.196i)6-s + (0.415 + 1.55i)7-s + 0.353·8-s + (0.731 − 0.422i)9-s + (−0.647 + 0.285i)10-s − 0.646i·11-s + (0.190 + 0.0509i)12-s + (0.714 − 1.23i)13-s + (0.802 − 0.802i)14-s + (−0.389 + 0.0600i)15-s + (−0.125 − 0.216i)16-s + (0.675 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.146 + 0.989i$
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: \((2,\ 370,\ (\ :1/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.764113 - 0.885550i\)
\(L(\frac12)\) \(\approx\) \(0.764113 - 0.885550i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.242 + 2.22i)T \)
37 \( 1 + (-2.82 - 5.38i)T \)
good3 \( 1 + (0.176 + 0.658i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.09 - 4.10i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.14iT - 11T^{2} \)
13 \( 1 + (-2.57 + 4.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.78 + 1.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.69 - 1.79i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.23T + 23T^{2} \)
29 \( 1 + (-0.485 + 0.485i)T - 29iT^{2} \)
31 \( 1 + (5.21 + 5.21i)T + 31iT^{2} \)
41 \( 1 + (-0.247 - 0.142i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + (-2.42 + 2.42i)T - 47iT^{2} \)
53 \( 1 + (-1.93 + 7.23i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.59 - 5.94i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.45 + 1.72i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.37 + 0.635i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.41 - 11.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.49 - 9.49i)T - 73iT^{2} \)
79 \( 1 + (-0.585 + 0.156i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.951 - 3.55i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.82 - 1.56i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 14.2iT - 97T^{2} \)
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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

−11.32352027959962317098403269178, −10.20136274600172414514157254794, −9.160165878484345965906556868395, −8.524071163591879016912143096801, −7.81027839385176972287203544849, −6.07699209650845027284078120608, −5.37502716937056175409847643596, −3.95282146785371289514597840079, −2.40632518778476832080171068520, −1.03072640059363434706102487330, 1.72231556239540651115924893827, 3.93590317825874163599101634835, 4.53613197076115745149847063743, 6.16419153896224229954185907482, 7.17840445945400035937374509628, 7.47462485139777652025019653888, 8.924660036570524370418447543886, 9.964559904646434047878249409184, 10.83983620274391753372329844496, 10.91291683028528591279926583584

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