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Properties

Label 2-370-185.103-c1-0-12
Degree $2$
Conductor $370$
Sign $0.572 + 0.819i$
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 + (0.434 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (1.34 − 1.78i)5-s + (1.18 − 1.18i)6-s + (−0.632 − 2.36i)7-s + 0.999·8-s + (0.159 − 0.0920i)9-s + (−2.21 − 0.274i)10-s − 3.49i·11-s + (−1.62 − 0.434i)12-s + (−1.48 + 2.56i)13-s + (−1.72 + 1.72i)14-s + (3.47 + 1.40i)15-s + (−0.5 − 0.866i)16-s + (1.91 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 + 0.935i)3-s + (−0.249 + 0.433i)4-s + (0.602 − 0.798i)5-s + (0.484 − 0.484i)6-s + (−0.239 − 0.892i)7-s + 0.353·8-s + (0.0531 − 0.0306i)9-s + (−0.701 − 0.0866i)10-s − 1.05i·11-s + (−0.467 − 0.125i)12-s + (−0.411 + 0.712i)13-s + (−0.462 + 0.462i)14-s + (0.898 + 0.363i)15-s + (−0.125 − 0.216i)16-s + (0.464 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.572 + 0.819i$
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: \((2,\ 370,\ (\ :1/2),\ 0.572 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15585 - 0.602316i\)
\(L(\frac12)\) \(\approx\) \(1.15585 - 0.602316i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 + 1.78i)T \)
37 \( 1 + (-5.95 + 1.25i)T \)
good3 \( 1 + (-0.434 - 1.62i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.632 + 2.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 3.49iT - 11T^{2} \)
13 \( 1 + (1.48 - 2.56i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.91 + 1.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.37 + 0.367i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.35T + 23T^{2} \)
29 \( 1 + (-2.69 + 2.69i)T - 29iT^{2} \)
31 \( 1 + (-1.18 - 1.18i)T + 31iT^{2} \)
41 \( 1 + (-2.80 - 1.61i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.95T + 43T^{2} \)
47 \( 1 + (7.46 - 7.46i)T - 47iT^{2} \)
53 \( 1 + (2.23 - 8.34i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.795 + 2.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.62 - 0.972i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.62 - 2.04i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.78 + 4.81i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.28 + 6.28i)T - 73iT^{2} \)
79 \( 1 + (14.7 - 3.96i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (2.69 - 10.0i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.87 - 2.11i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−11.03796863682656964052910006854, −10.14092420476034550409964443665, −9.554388965491508442830191305922, −8.894969890255218375563066625624, −7.80479173994171996868426906247, −6.44133072278064947631924761459, −4.98403216669089581033699694482, −4.14863106259195351809192265796, −3.01624675996676861498514405569, −1.09657019145110161093586511413, 1.77505759198463330078897552248, 2.91118011217007532789242216703, 4.99740689577046792175691146950, 6.05458167382555620364238933143, 6.92664898741843367060212817334, 7.59148900048829736443579047261, 8.575509155584551179024433351533, 9.796334025695685596006800098819, 10.21325713440623753226378825980, 11.64259007753022259423602534779

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