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Properties

Label 2-315-45.34-c1-0-18
Degree $2$
Conductor $315$
Sign $-0.232 + 0.972i$
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  + (−0.866 − 0.5i)2-s − 1.73·3-s + (−0.500 − 0.866i)4-s + (1.86 + 1.23i)5-s + (1.49 + 0.866i)6-s + (−0.866 − 0.5i)7-s + 3i·8-s + 2.99·9-s + (−1 − 2i)10-s + (1.5 − 2.59i)11-s + (0.866 + 1.49i)12-s + (0.866 − 0.5i)13-s + (0.499 + 0.866i)14-s + (−3.23 − 2.13i)15-s + (0.500 − 0.866i)16-s − 7i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s − 1.00·3-s + (−0.250 − 0.433i)4-s + (0.834 + 0.550i)5-s + (0.612 + 0.353i)6-s + (−0.327 − 0.188i)7-s + 1.06i·8-s + 0.999·9-s + (−0.316 − 0.632i)10-s + (0.452 − 0.783i)11-s + (0.250 + 0.433i)12-s + (0.240 − 0.138i)13-s + (0.133 + 0.231i)14-s + (−0.834 − 0.550i)15-s + (0.125 − 0.216i)16-s − 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.232 + 0.972i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.232 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381264 - 0.483068i\)
\(L(\frac12)\) \(\approx\) \(0.381264 - 0.483068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 + (-1.86 - 1.23i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.79 - 4.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + 3iT - 73T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.33 + 2.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (4.33 + 2.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.05516294274860558872055561476, −10.57836344672918235032261326426, −9.651184422703226285169379244333, −9.019995820868694918559785398913, −7.44646811626751569166101267025, −6.20464139653574160431434222451, −5.73753709263598194581812969663, −4.35820986211682045510658817944, −2.38324604528608972742369904198, −0.65912522498567746488438829373, 1.55482727834239999611678247348, 3.93387277418203653643475218529, 5.01148511360456842377543932484, 6.35333956850444061378629582218, 6.81884260522802219274664819798, 8.357981900533143131951681217761, 9.041367744333648203797976083333, 10.08572923872973721075690474022, 10.65968226960591735998604050243, 12.36533305466716200343859333112

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