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Properties

Label 2-370-185.97-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.884 + 0.466i$
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 + (−0.241 + 0.900i)3-s + (−0.499 − 0.866i)4-s + (−2.06 + 0.851i)5-s + (−0.658 − 0.658i)6-s + (−0.406 + 1.51i)7-s + 0.999·8-s + (1.84 + 1.06i)9-s + (0.296 − 2.21i)10-s + 2.93i·11-s + (0.900 − 0.241i)12-s + (−1.76 − 3.04i)13-s + (−1.11 − 1.11i)14-s + (−0.267 − 2.06i)15-s + (−0.5 + 0.866i)16-s + (−6.36 − 3.67i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.139 + 0.519i)3-s + (−0.249 − 0.433i)4-s + (−0.924 + 0.380i)5-s + (−0.269 − 0.269i)6-s + (−0.153 + 0.573i)7-s + 0.353·8-s + (0.615 + 0.355i)9-s + (0.0938 − 0.700i)10-s + 0.884i·11-s + (0.259 − 0.0696i)12-s + (−0.488 − 0.845i)13-s + (−0.297 − 0.297i)14-s + (−0.0690 − 0.533i)15-s + (−0.125 + 0.216i)16-s + (−1.54 − 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0973086 - 0.393402i\)
\(L(\frac12)\) \(\approx\) \(0.0973086 - 0.393402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 - 0.851i)T \)
37 \( 1 + (2.51 - 5.53i)T \)
good3 \( 1 + (0.241 - 0.900i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.406 - 1.51i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 2.93iT - 11T^{2} \)
13 \( 1 + (1.76 + 3.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.36 + 3.67i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.85 + 1.30i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.62T + 23T^{2} \)
29 \( 1 + (-4.48 - 4.48i)T + 29iT^{2} \)
31 \( 1 + (-3.51 + 3.51i)T - 31iT^{2} \)
41 \( 1 + (5.22 - 3.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 0.121T + 43T^{2} \)
47 \( 1 + (-6.92 - 6.92i)T + 47iT^{2} \)
53 \( 1 + (1.14 + 4.27i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.71 - 13.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (12.9 + 3.46i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-9.85 - 2.64i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.60 + 6.23i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.65 + 4.65i)T + 73iT^{2} \)
79 \( 1 + (-2.20 - 0.591i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.38 - 12.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.58 - 0.423i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−11.86873936397296544600787299088, −10.73684408702900614893084209808, −10.14354838240977547411095185086, −9.117718857986778992713753695938, −8.139537992623255229279348691302, −7.24532315102894271286044616556, −6.42307720369898090575056430328, −4.89420123397317284334478852936, −4.30097156660221451183841742666, −2.51029848366842316997094735258, 0.30284025260101515416172942380, 1.96553056301890632077740146468, 3.85961009064519221859888269404, 4.37589917349505044034413530156, 6.33225869520695041644744005004, 7.13638777376424208685075035852, 8.256504295135300687701304006149, 8.854725101002082332796947827681, 10.14813937814146098243992084076, 10.91344241496102743103659992800

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