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Properties

Label 2-315-63.58-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.607 + 0.794i$
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  − 1.69·2-s + (−1.59 − 0.671i)3-s + 0.857·4-s + (−0.5 − 0.866i)5-s + (2.69 + 1.13i)6-s + (2.40 − 1.10i)7-s + 1.93·8-s + (2.09 + 2.14i)9-s + (0.845 + 1.46i)10-s + (−2.27 + 3.93i)11-s + (−1.36 − 0.575i)12-s + (1.58 − 2.74i)13-s + (−4.06 + 1.85i)14-s + (0.216 + 1.71i)15-s − 4.97·16-s + (−2.83 − 4.91i)17-s + ⋯
L(s)  = 1  − 1.19·2-s + (−0.921 − 0.387i)3-s + 0.428·4-s + (−0.223 − 0.387i)5-s + (1.10 + 0.463i)6-s + (0.909 − 0.415i)7-s + 0.682·8-s + (0.699 + 0.714i)9-s + (0.267 + 0.462i)10-s + (−0.685 + 1.18i)11-s + (−0.395 − 0.166i)12-s + (0.439 − 0.761i)13-s + (−1.08 + 0.497i)14-s + (0.0559 + 0.443i)15-s − 1.24·16-s + (−0.687 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155529 - 0.314878i\)
\(L(\frac12)\) \(\approx\) \(0.155529 - 0.314878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 + 0.671i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.40 + 1.10i)T \)
good2 \( 1 + 1.69T + 2T^{2} \)
11 \( 1 + (2.27 - 3.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.58 + 2.74i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.83 + 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.70 + 2.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0116 - 0.0202i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.17 + 8.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.71T + 31T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.45 - 5.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.52 + 7.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 + (5.10 + 8.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.28T + 59T^{2} \)
61 \( 1 + 8.57T + 61T^{2} \)
67 \( 1 + 1.50T + 67T^{2} \)
71 \( 1 - 9.66T + 71T^{2} \)
73 \( 1 + (5.85 + 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.16T + 79T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.27 + 7.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.533 + 0.924i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25098000312961034029446296999, −10.33125977386348938810109552266, −9.616905273823361701630687299854, −8.295385440734761815411089642946, −7.63680885383232908497785022835, −6.89697954601191196499777795286, −5.14239449331220180543050864176, −4.57837445731341905436734545496, −1.90923276326416564298159961742, −0.44731372457148472122644739188, 1.56868998412803397733634823909, 3.82761666500149640075065228124, 5.11761509193772724789082801671, 6.18488567437006763680540076540, 7.39670369357128592063755933101, 8.432750705844555495802157533645, 9.063813150294862452217009417537, 10.35358458737523999429905504469, 10.95099485855963062467563622528, 11.38913918819782880120796413178

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