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Properties

Label 2-370-37.10-c1-0-8
Degree $2$
Conductor $370$
Sign $-0.389 + 0.921i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (−2.58 − 4.47i)7-s − 0.999·8-s + (1 − 1.73i)9-s − 0.999·10-s − 1.16·11-s + (0.499 − 0.866i)12-s + (2.08 + 3.60i)13-s − 5.16·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.581 + 1.00i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (−0.975 − 1.68i)7-s − 0.353·8-s + (0.333 − 0.577i)9-s − 0.316·10-s − 0.350·11-s + (0.144 − 0.249i)12-s + (0.577 + 0.999i)13-s − 1.37·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.140 + 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771133 - 1.16275i\)
\(L(\frac12)\) \(\approx\) \(0.771133 - 1.16275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (6.08 + 0.140i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.58 + 4.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-2.08 - 3.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.581 - 1.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.16 + 7.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 4.16T + 31T^{2} \)
41 \( 1 + (2.66 + 4.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 - 5.16T + 47T^{2} \)
53 \( 1 + (7.08 - 12.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.581 + 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 7.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.16 - 7.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.16 + 5.47i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.48T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.32 + 14.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−10.82074690195222276451661557181, −10.42167334909846941011005694818, −9.371074288863706119755890118952, −8.718888934810170215192977108381, −7.07530387084052659853316631830, −6.45932592562941152013652533914, −4.53189737304267864144303863949, −4.13064452592503863477750881196, −3.00546221648908449843011707036, −0.846004990505883787091900343162, 2.41594353827370523126456088029, 3.37243343721468402506650079847, 5.06752791065000670970333675116, 6.06187629044709327909016780517, 6.75523878169826539666907379096, 8.159175167445902562343513687573, 8.415203042153325915714280403408, 9.782281948823175082458951043221, 10.72199561467672104050707787259, 12.16027824726497892574224771079

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