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Properties

Label 2-315-21.20-c1-0-5
Degree $2$
Conductor $315$
Sign $0.537 + 0.843i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  − 0.517i·2-s + 1.73·4-s − 5-s + (1 − 2.44i)7-s − 1.93i·8-s + 0.517i·10-s + 0.378i·11-s − 1.79i·13-s + (−1.26 − 0.517i)14-s + 2.46·16-s + 3.46·17-s + 1.79i·19-s − 1.73·20-s + 0.196·22-s − 1.41i·23-s + ⋯
L(s)  = 1  − 0.366i·2-s + 0.866·4-s − 0.447·5-s + (0.377 − 0.925i)7-s − 0.683i·8-s + 0.163i·10-s + 0.114i·11-s − 0.497i·13-s + (−0.338 − 0.138i)14-s + 0.616·16-s + 0.840·17-s + 0.411i·19-s − 0.387·20-s + 0.0418·22-s − 0.294i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35109 - 0.740804i\)
\(L(\frac12)\) \(\approx\) \(1.35109 - 0.740804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (-1 + 2.44i)T \)
good2 \( 1 + 0.517iT - 2T^{2} \)
11 \( 1 - 0.378iT - 11T^{2} \)
13 \( 1 + 1.79iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 1.79iT - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 6.69iT - 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 9.46T + 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 - 1.79iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 9.46T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−11.58170600300080260044800082087, −10.42118838702473500227668003705, −10.22029462632410244121665066787, −8.529520374171114882506815604666, −7.54843483348540299013991440282, −6.89055431914241155868229891963, −5.54077156044788334666290295041, −4.09896517338678989302447389888, −3.02764358493420422960875468139, −1.30303491816795261239176676087, 1.97407916811687505288819705787, 3.32965442936925613306627543020, 4.99995413466858911931645253680, 5.95927675528890213325119011227, 7.00660177259172093209196409977, 7.942762985979819311952952182424, 8.758946043484553395388308297337, 9.970196009090520321184328461966, 11.16510169441811270540499428614, 11.73283931706842662925422231673

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