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Properties

Label 2-315-63.20-c1-0-15
Degree $2$
Conductor $315$
Sign $0.151 - 0.988i$
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  + (1.58 + 0.912i)2-s + (0.791 + 1.54i)3-s + (0.666 + 1.15i)4-s + (0.5 + 0.866i)5-s + (−0.155 + 3.15i)6-s + (2.11 − 1.58i)7-s − 1.21i·8-s + (−1.74 + 2.43i)9-s + 1.82i·10-s + (−0.549 − 0.317i)11-s + (−1.25 + 1.93i)12-s + (−2.73 + 1.57i)13-s + (4.79 − 0.580i)14-s + (−0.938 + 1.45i)15-s + (2.44 − 4.23i)16-s − 2.83·17-s + ⋯
L(s)  = 1  + (1.11 + 0.645i)2-s + (0.456 + 0.889i)3-s + (0.333 + 0.576i)4-s + (0.223 + 0.387i)5-s + (−0.0635 + 1.28i)6-s + (0.799 − 0.600i)7-s − 0.430i·8-s + (−0.582 + 0.812i)9-s + 0.577i·10-s + (−0.165 − 0.0956i)11-s + (−0.361 + 0.559i)12-s + (−0.758 + 0.437i)13-s + (1.28 − 0.155i)14-s + (−0.242 + 0.375i)15-s + (0.611 − 1.05i)16-s − 0.686·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99048 + 1.70778i\)
\(L(\frac12)\) \(\approx\) \(1.99048 + 1.70778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.791 - 1.54i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.11 + 1.58i)T \)
good2 \( 1 + (-1.58 - 0.912i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (0.549 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.73 - 1.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.83T + 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + (3.83 - 2.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0996 - 0.0575i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.272 + 0.157i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + (1.11 + 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.870 - 1.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.95 + 6.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.63iT - 53T^{2} \)
59 \( 1 + (-7.65 - 13.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.94 + 5.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 + 4.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 6.45iT - 73T^{2} \)
79 \( 1 + (-6.51 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.25 + 3.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + (-0.266 - 0.154i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.86164817470256981795997726395, −10.94732204534008873734401007661, −10.03793880204760056995038188080, −9.126427869269412768405349377667, −7.80621171874456894526029205715, −6.94718034615244116472906181008, −5.66837553289012141055545925901, −4.67348557850352200260106062217, −4.05971633735233227396405916821, −2.58887372892133672754862341314, 1.84607243581455553144095884434, 2.72301662004232570093141470473, 4.21858498962850663065805699377, 5.32043432741861869085449937172, 6.22626200375244510830665011282, 7.84763269820929457010526673952, 8.352636384014141779064701198804, 9.589420607264876524969924798052, 10.95242219095626195378729432474, 11.90772207670958803990379093987

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