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Properties

Label 2-385-7.4-c1-0-20
Degree $2$
Conductor $385$
Sign $-0.525 + 0.850i$
Analytic cond. $3.07424$
Root an. cond. $1.75335$
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  + (−0.461 − 0.800i)2-s + (1.05 − 1.82i)3-s + (0.573 − 0.992i)4-s + (0.5 + 0.866i)5-s − 1.94·6-s + (2.24 − 1.39i)7-s − 2.90·8-s + (−0.719 − 1.24i)9-s + (0.461 − 0.800i)10-s + (0.5 − 0.866i)11-s + (−1.20 − 2.09i)12-s + 2.69·13-s + (−2.15 − 1.15i)14-s + 2.10·15-s + (0.196 + 0.340i)16-s + (−2.29 + 3.97i)17-s + ⋯
L(s)  = 1  + (−0.326 − 0.565i)2-s + (0.608 − 1.05i)3-s + (0.286 − 0.496i)4-s + (0.223 + 0.387i)5-s − 0.794·6-s + (0.848 − 0.528i)7-s − 1.02·8-s + (−0.239 − 0.415i)9-s + (0.146 − 0.253i)10-s + (0.150 − 0.261i)11-s + (−0.348 − 0.603i)12-s + 0.746·13-s + (−0.576 − 0.307i)14-s + 0.543·15-s + (0.0491 + 0.0851i)16-s + (−0.557 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(3.07424\)
Root analytic conductor: \(1.75335\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797263 - 1.43012i\)
\(L(\frac12)\) \(\approx\) \(0.797263 - 1.43012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.24 + 1.39i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.461 + 0.800i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.05 + 1.82i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 + (2.29 - 3.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.40 - 5.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.78T + 29T^{2} \)
31 \( 1 + (4.35 - 7.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.58 + 2.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 1.01T + 43T^{2} \)
47 \( 1 + (-3.18 - 5.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.54 + 6.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.30 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.94 - 8.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + (3.39 - 5.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.09 + 8.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 + (-2.43 - 4.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.44T + 97T^{2} \)
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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

−10.98021452064354692116836351225, −10.41597166576226922866589168019, −9.017205911874170358696584408467, −8.415163590880455111794482026339, −7.17046099537504313887835533699, −6.57936417844749773350064872792, −5.27760125192878415932354563300, −3.51281570475809511296742993829, −2.12669785374093018264247316575, −1.32713477382990628853693874071, 2.28207732029984982822476115865, 3.66738585427080071169848242313, 4.69021482970629538162124654130, 5.89479106868509827903625403597, 7.07950752722692419919524533841, 8.343242399752376850685889257752, 8.704739439581895310462532914444, 9.487882597242823913034598782632, 10.60656479088485090485280136821, 11.58627095993524834518462000347

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