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Properties

Label 2-315-63.25-c1-0-25
Degree $2$
Conductor $315$
Sign $0.900 + 0.435i$
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  + 2.58·2-s + (−1.01 − 1.40i)3-s + 4.68·4-s + (−0.5 + 0.866i)5-s + (−2.62 − 3.62i)6-s + (2.41 + 1.07i)7-s + 6.94·8-s + (−0.938 + 2.84i)9-s + (−1.29 + 2.23i)10-s + (−1.23 − 2.13i)11-s + (−4.75 − 6.57i)12-s + (−3.19 − 5.52i)13-s + (6.25 + 2.77i)14-s + (1.72 − 0.177i)15-s + 8.58·16-s + (−2.57 + 4.45i)17-s + ⋯
L(s)  = 1  + 1.82·2-s + (−0.586 − 0.810i)3-s + 2.34·4-s + (−0.223 + 0.387i)5-s + (−1.07 − 1.48i)6-s + (0.913 + 0.406i)7-s + 2.45·8-s + (−0.312 + 0.949i)9-s + (−0.408 + 0.708i)10-s + (−0.372 − 0.645i)11-s + (−1.37 − 1.89i)12-s + (−0.885 − 1.53i)13-s + (1.67 + 0.742i)14-s + (0.444 − 0.0458i)15-s + 2.14·16-s + (−0.623 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93664 - 0.673408i\)
\(L(\frac12)\) \(\approx\) \(2.93664 - 0.673408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.01 + 1.40i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.41 - 1.07i)T \)
good2 \( 1 - 2.58T + 2T^{2} \)
11 \( 1 + (1.23 + 2.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.19 + 5.52i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.57 - 4.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.825 - 1.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.42 - 4.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.321 + 0.557i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.547T + 31T^{2} \)
37 \( 1 + (3.30 + 5.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.42 - 7.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.75 - 6.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.63T + 47T^{2} \)
53 \( 1 + (-4.66 + 8.07i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.89T + 59T^{2} \)
61 \( 1 + 0.314T + 61T^{2} \)
67 \( 1 + 1.18T + 67T^{2} \)
71 \( 1 + 1.21T + 71T^{2} \)
73 \( 1 + (-6.30 + 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 + (-3.21 + 5.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.44 + 2.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.53 + 9.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.83558546361690968701491550366, −11.12265110133837709678800079832, −10.45571053203193085689969994737, −8.076059930997984047473531565570, −7.56122377446679059966281017068, −6.23902538998601131019255650533, −5.59297315702680789630616388169, −4.75779743763476318339115808800, −3.22151049264140814338551583357, −2.03643723286142206928289264685, 2.30682746595608371700116689333, 4.02357880900464846936609983944, 4.71762252219114162860806067009, 5.13082085280769407835608106705, 6.60360503891658154166866795356, 7.35893717958569061768077972044, 9.042605201923400622108699658944, 10.30439785892366444178654865261, 11.23623833111091003791971876114, 11.90316845480256125559573985023

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