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Properties

Label 2-315-63.41-c1-0-14
Degree $2$
Conductor $315$
Sign $0.648 - 0.760i$
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  + (−1.41 + 0.817i)2-s + (1.73 − 0.0392i)3-s + (0.335 − 0.581i)4-s + (−0.5 + 0.866i)5-s + (−2.41 + 1.47i)6-s + (2.11 − 1.59i)7-s − 2.17i·8-s + (2.99 − 0.135i)9-s − 1.63i·10-s + (1.25 − 0.727i)11-s + (0.558 − 1.02i)12-s + (4.17 + 2.40i)13-s + (−1.69 + 3.97i)14-s + (−0.831 + 1.51i)15-s + (2.44 + 4.23i)16-s − 6.37·17-s + ⋯
L(s)  = 1  + (−1.00 + 0.577i)2-s + (0.999 − 0.0226i)3-s + (0.167 − 0.290i)4-s + (−0.223 + 0.387i)5-s + (−0.987 + 0.600i)6-s + (0.799 − 0.601i)7-s − 0.767i·8-s + (0.998 − 0.0453i)9-s − 0.516i·10-s + (0.379 − 0.219i)11-s + (0.161 − 0.294i)12-s + (1.15 + 0.667i)13-s + (−0.452 + 1.06i)14-s + (−0.214 + 0.392i)15-s + (0.611 + 1.05i)16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.648 - 0.760i$
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: \((2,\ 315,\ (\ :1/2),\ 0.648 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04836 + 0.483849i\)
\(L(\frac12)\) \(\approx\) \(1.04836 + 0.483849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0392i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.11 + 1.59i)T \)
good2 \( 1 + (1.41 - 0.817i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.25 + 0.727i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.17 - 2.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.37T + 17T^{2} \)
19 \( 1 + 1.12iT - 19T^{2} \)
23 \( 1 + (-7.62 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.63 - 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.94 + 4.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.26T + 37T^{2} \)
41 \( 1 + (4.21 - 7.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.739 - 1.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.46iT - 53T^{2} \)
59 \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.44 - 3.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.282 - 0.489i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.7iT - 71T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + (3.80 + 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.62 + 2.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.18T + 89T^{2} \)
97 \( 1 + (10.0 - 5.79i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.32070433305626539608005451219, −10.83613847306293172584958824792, −9.378872953967566584667597996273, −8.959724439224289725815619888608, −8.046970995838694558802028079052, −7.24449771686460061064042723860, −6.52994287671617507939901024735, −4.41979313369246863777745265329, −3.48165532877687061341189830046, −1.52066424611898864503498390070, 1.39271950486070566728003798533, 2.56178042569479696362676781450, 4.15530413411441700400489348017, 5.41663013249591332547467569650, 7.12717597689282506336509842080, 8.320117536924214821245588505727, 8.775298073800707513698226966322, 9.289466698096083478726822424006, 10.69334069408275519324816837024, 11.10992002061899351440778541162

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