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Properties

Label 2-315-63.41-c1-0-3
Degree $2$
Conductor $315$
Sign $0.398 - 0.917i$
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  + (0.963 − 0.556i)2-s + (−1.73 − 0.0368i)3-s + (−0.381 + 0.660i)4-s + (0.5 − 0.866i)5-s + (−1.68 + 0.927i)6-s + (−1.36 + 2.26i)7-s + 3.07i·8-s + (2.99 + 0.127i)9-s − 1.11i·10-s + (−2.20 + 1.27i)11-s + (0.684 − 1.12i)12-s + (5.31 + 3.07i)13-s + (−0.0525 + 2.94i)14-s + (−0.897 + 1.48i)15-s + (0.946 + 1.63i)16-s − 3.62·17-s + ⋯
L(s)  = 1  + (0.681 − 0.393i)2-s + (−0.999 − 0.0212i)3-s + (−0.190 + 0.330i)4-s + (0.223 − 0.387i)5-s + (−0.689 + 0.378i)6-s + (−0.515 + 0.856i)7-s + 1.08i·8-s + (0.999 + 0.0424i)9-s − 0.351i·10-s + (−0.663 + 0.383i)11-s + (0.197 − 0.326i)12-s + (1.47 + 0.851i)13-s + (−0.0140 + 0.786i)14-s + (−0.231 + 0.382i)15-s + (0.236 + 0.409i)16-s − 0.878·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.398 - 0.917i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921290 + 0.604495i\)
\(L(\frac12)\) \(\approx\) \(0.921290 + 0.604495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0368i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.36 - 2.26i)T \)
good2 \( 1 + (-0.963 + 0.556i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.20 - 1.27i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.31 - 3.07i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
19 \( 1 - 5.85iT - 19T^{2} \)
23 \( 1 + (0.351 + 0.202i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.74 + 4.47i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (-2.20 + 3.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.15 + 1.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.93 - 3.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (-5.15 + 8.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.26 + 2.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 7.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 - 2.90iT - 73T^{2} \)
79 \( 1 + (-5.76 - 9.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.99 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (9.04 - 5.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.13888174368568007430544444440, −11.17227799707114605195944933073, −10.22475129024490350294214208726, −9.010871013288137166440288476957, −8.167811995107260965788704209010, −6.56868369198922264304922281417, −5.75498841270350100364559161547, −4.77482694278698571876077201243, −3.74604414206081240546554045917, −2.04611699523433436942788726312, 0.75329801480028926004509726863, 3.40091660726908182055360347100, 4.60139414792992099729664393267, 5.56447989730415630330359488783, 6.50067364944639360876174842915, 7.02002218032862749828759593627, 8.657049441834769220407454364210, 10.07231946837958720115883058922, 10.56270402944049536666268504838, 11.28982138796491442758636713709

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