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Properties

Label 2-315-63.25-c1-0-7
Degree $2$
Conductor $315$
Sign $0.0630 - 0.998i$
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  − 1.03·2-s + (−0.388 + 1.68i)3-s − 0.927·4-s + (0.5 − 0.866i)5-s + (0.402 − 1.74i)6-s + (2.63 − 0.275i)7-s + 3.03·8-s + (−2.69 − 1.31i)9-s + (−0.517 + 0.896i)10-s + (2.91 + 5.05i)11-s + (0.360 − 1.56i)12-s + (−1.79 − 3.10i)13-s + (−2.72 + 0.285i)14-s + (1.26 + 1.18i)15-s − 1.28·16-s + (−1.70 + 2.96i)17-s + ⋯
L(s)  = 1  − 0.732·2-s + (−0.224 + 0.974i)3-s − 0.463·4-s + (0.223 − 0.387i)5-s + (0.164 − 0.713i)6-s + (0.994 − 0.104i)7-s + 1.07·8-s + (−0.899 − 0.437i)9-s + (−0.163 + 0.283i)10-s + (0.880 + 1.52i)11-s + (0.104 − 0.451i)12-s + (−0.497 − 0.861i)13-s + (−0.728 + 0.0761i)14-s + (0.327 + 0.304i)15-s − 0.321·16-s + (−0.414 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0630 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0630 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0630 - 0.998i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0630 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569902 + 0.535057i\)
\(L(\frac12)\) \(\approx\) \(0.569902 + 0.535057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.388 - 1.68i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.63 + 0.275i)T \)
good2 \( 1 + 1.03T + 2T^{2} \)
11 \( 1 + (-2.91 - 5.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.79 + 3.10i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.70 - 2.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.47 - 4.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.28 - 5.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.65 - 2.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.15T + 31T^{2} \)
37 \( 1 + (-0.0816 - 0.141i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.52 - 7.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.873 + 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 + (-3.06 + 5.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.19T + 59T^{2} \)
61 \( 1 - 4.83T + 61T^{2} \)
67 \( 1 + 16.2T + 67T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + (-5.42 + 9.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.06T + 79T^{2} \)
83 \( 1 + (-0.158 + 0.274i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.01 - 1.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.865 - 1.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.73489493172801619014186580654, −10.58778938209237509074997185422, −9.889543141444842391930627087632, −9.300205080669145626263432466354, −8.281079177275092023301728008785, −7.46109249321023258934143483370, −5.67864975372476160248656254536, −4.73386566161907073442574805484, −3.97485887779254905792925079020, −1.59323153498889802959578609276, 0.842090568467256725465137673495, 2.37317977759156083849702936817, 4.36706099102958991982969973672, 5.63582900578772289495553217674, 6.79903580157201617356910932942, 7.68712437092024951655538934297, 8.691221635759542498139663395121, 9.197535167190989787771726929932, 10.68910704676078464947194012855, 11.37712173321738402530049069980

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