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Properties

Label 2-315-7.2-c3-0-13
Degree $2$
Conductor $315$
Sign $-0.576 - 0.817i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.21 + 3.82i)2-s + (−5.77 − 9.99i)4-s + (2.5 − 4.33i)5-s + (0.502 + 18.5i)7-s + 15.6·8-s + (11.0 + 19.1i)10-s + (−20.8 − 36.0i)11-s + 78.9·13-s + (−71.9 − 38.9i)14-s + (11.5 − 20.0i)16-s + (17.2 + 29.9i)17-s + (15.7 − 27.2i)19-s − 57.7·20-s + 184.·22-s + (−1.52 + 2.64i)23-s + ⋯
L(s)  = 1  + (−0.781 + 1.35i)2-s + (−0.721 − 1.24i)4-s + (0.223 − 0.387i)5-s + (0.0271 + 0.999i)7-s + 0.692·8-s + (0.349 + 0.605i)10-s + (−0.571 − 0.989i)11-s + 1.68·13-s + (−1.37 − 0.744i)14-s + (0.180 − 0.312i)16-s + (0.246 + 0.427i)17-s + (0.190 − 0.329i)19-s − 0.645·20-s + 1.78·22-s + (−0.0138 + 0.0239i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.576 - 0.817i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -0.576 - 0.817i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.166108811\)
\(L(\frac12)\) \(\approx\) \(1.166108811\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-0.502 - 18.5i)T \)
good2 \( 1 + (2.21 - 3.82i)T + (-4 - 6.92i)T^{2} \)
11 \( 1 + (20.8 + 36.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 78.9T + 2.19e3T^{2} \)
17 \( 1 + (-17.2 - 29.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-15.7 + 27.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (1.52 - 2.64i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 87.7T + 2.43e4T^{2} \)
31 \( 1 + (-137. - 238. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (141. - 245. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 57.3T + 6.89e4T^{2} \)
43 \( 1 - 502.T + 7.95e4T^{2} \)
47 \( 1 + (235. - 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (341. + 590. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-277. - 481. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-80.1 + 138. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (33.6 + 58.2i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 221.T + 3.57e5T^{2} \)
73 \( 1 + (-118. - 205. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (237. - 411. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + (462. - 800. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 144.T + 9.12e5T^{2} \)
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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

−11.41132520645164379935785026115, −10.35577192324140546709246552229, −9.195263318418313728613119448814, −8.485547312040184563211161825086, −8.113273125152184894550985330088, −6.53999125531942569918701622413, −5.93509638005296884722104156300, −5.07738754230489259004950361021, −3.13456566811974018003686911760, −1.07251152355150579725444917541, 0.71041083133130910127692564688, 1.95088094110017351597698234354, 3.27453719750122303705298236366, 4.28709891541823680233278842563, 6.02433483321733464549374801889, 7.33438731506085754577184714202, 8.260582769377244625988582980190, 9.398986660480264802370871341129, 10.18020254706212585269170548123, 10.76072970319568468457722362585

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