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Properties

Label 2-370-37.16-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.222 - 0.974i$
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  + (−0.939 − 0.342i)2-s + (−3.13 + 1.14i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 3.33·6-s + (0.599 − 3.39i)7-s + (−0.500 − 0.866i)8-s + (6.21 − 5.21i)9-s + (0.5 − 0.866i)10-s + (2.11 + 3.65i)11-s + (−3.13 − 1.14i)12-s + (0.742 + 0.623i)13-s + (−1.72 + 2.98i)14-s + (−0.578 − 3.28i)15-s + (0.173 + 0.984i)16-s + (−0.826 + 0.693i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−1.80 + 0.658i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + 1.36·6-s + (0.226 − 1.28i)7-s + (−0.176 − 0.306i)8-s + (2.07 − 1.73i)9-s + (0.158 − 0.273i)10-s + (0.636 + 1.10i)11-s + (−0.904 − 0.329i)12-s + (0.206 + 0.172i)13-s + (−0.461 + 0.798i)14-s + (−0.149 − 0.847i)15-s + (0.0434 + 0.246i)16-s + (−0.200 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.222 - 0.974i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.222 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246335 + 0.308885i\)
\(L(\frac12)\) \(\approx\) \(0.246335 + 0.308885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-4.62 - 3.94i)T \)
good3 \( 1 + (3.13 - 1.14i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.599 + 3.39i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.11 - 3.65i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.742 - 0.623i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.826 - 0.693i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (5.68 - 2.06i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (1.24 - 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.847 - 1.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
41 \( 1 + (-7.33 - 6.15i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 2.41T + 43T^{2} \)
47 \( 1 + (3.34 - 5.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.74 - 9.91i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.88 + 10.7i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.46 + 7.94i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.54 - 14.4i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.75 + 1.36i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 8.09T + 73T^{2} \)
79 \( 1 + (2.12 - 12.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.02 - 0.861i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-2.58 - 14.6i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.61 + 7.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.20767974278706091805031450072, −10.85502916109799108261967044917, −10.07724321778525433408290561301, −9.415467906714845554210730234203, −7.67301884649706677230936871576, −6.80006284220039504572572069010, −6.13015675884099081670906546136, −4.51756176259271652006496843895, −3.97735100439555166318905608868, −1.35073249125834919530736833094, 0.45792955445783591623024673377, 2.00891390667242521148361333199, 4.56842338225961513400475777735, 5.86345805709580585094501353733, 5.99969459865560796151225963619, 7.19611690753291596170811019701, 8.391681051086211214354717049344, 9.127956642594464507466600352704, 10.55426028031518390836543452180, 11.20157137377893752383968266943

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