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Properties

Label 1-315-315.2-r0-0-0
Degree $1$
Conductor $315$
Sign $0.0810 + 0.996i$
Analytic cond. $1.46285$
Root an. cond. $1.46285$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s − 11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s + (0.5 + 0.866i)26-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s + (0.5 + 0.866i)34-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s − 11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + i·23-s + (0.5 + 0.866i)26-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (−0.866 + 0.5i)32-s + (0.5 + 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0810 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0810 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0810 + 0.996i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ 0.0810 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.490327290 + 1.373997389i\)
\(L(\frac12)\) \(\approx\) \(1.490327290 + 1.373997389i\)
\(L(1)\) \(\approx\) \(1.493167676 + 0.7334537779i\)
\(L(1)\) \(\approx\) \(1.493167676 + 0.7334537779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.88166869244542005744942085594, −23.73781298151104947864038365044, −23.28111109709894957623147276272, −22.2785288252883453733861680929, −21.4324539423338098347181175208, −20.53090839961584545082478549406, −19.99999874704623759741283187186, −18.60838227105662489505120877204, −18.203781326781820760489276426629, −16.51118442749048094716133173280, −15.75405731297739299621721578394, −14.85163525517572681042306414625, −13.826397286500396708077194959336, −13.05553004928340603171753499684, −12.222827861490589244651118926138, −11.06534457045552217075906462081, −10.43720384152167238366714742911, −9.27927548316793556663513422065, −7.901646666499579933410994288186, −6.74332006284773424739924657059, −5.568347257142444938472854590922, −4.82678660892252537050450604399, −3.43398331060666284448219304768, −2.61605528809912808037770800262, −1.07102682847878249423540236871, 1.8522111221584263968900096074, 3.24789557255859349030344901343, 4.13678701648250810969131495746, 5.489411996186308691359529749171, 6.075003933217442326048307088338, 7.520988310537710832371947632985, 8.07534652474417689591140886979, 9.49250196129752340354066845425, 10.814272398249090754750854643161, 11.71517662862577335488749500640, 12.76500078145093859216354777474, 13.53098046361563241432825702773, 14.41768736102232612588679386333, 15.41289398182911468730847453678, 16.175562635977390036278877979675, 16.998388178625800402902432558292, 18.13891942855315244575875156791, 19.059727967909920455315653508016, 20.480806671064369295830331329558, 21.03446117475551940159661314108, 21.85493252836609655797649139333, 23.01123798576223189669983839890, 23.506640545115943602898525877997, 24.3488355437276250065185271177, 25.43832565956505282061382853277

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