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Properties

Label 2-315-63.38-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.0126 + 0.999i$
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  − 2.34i·2-s + (−0.0139 + 1.73i)3-s − 3.51·4-s + (−0.5 + 0.866i)5-s + (4.06 + 0.0326i)6-s + (2.22 − 1.43i)7-s + 3.55i·8-s + (−2.99 − 0.0482i)9-s + (2.03 + 1.17i)10-s + (3.99 − 2.30i)11-s + (0.0488 − 6.08i)12-s + (5.08 − 2.93i)13-s + (−3.37 − 5.21i)14-s + (−1.49 − 0.878i)15-s + 1.31·16-s + (−2.10 + 3.64i)17-s + ⋯
L(s)  = 1  − 1.66i·2-s + (−0.00803 + 0.999i)3-s − 1.75·4-s + (−0.223 + 0.387i)5-s + (1.66 + 0.0133i)6-s + (0.839 − 0.543i)7-s + 1.25i·8-s + (−0.999 − 0.0160i)9-s + (0.643 + 0.371i)10-s + (1.20 − 0.695i)11-s + (0.0141 − 1.75i)12-s + (1.40 − 0.813i)13-s + (−0.902 − 1.39i)14-s + (−0.385 − 0.226i)15-s + 0.328·16-s + (−0.510 + 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0126 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0126 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0126 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0126 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924455 - 0.936186i\)
\(L(\frac12)\) \(\approx\) \(0.924455 - 0.936186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0139 - 1.73i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.22 + 1.43i)T \)
good2 \( 1 + 2.34iT - 2T^{2} \)
11 \( 1 + (-3.99 + 2.30i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.08 + 2.93i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.10 - 3.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.16 + 3.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.53 + 0.883i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.80 + 1.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.45iT - 31T^{2} \)
37 \( 1 + (-3.23 - 5.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.641 - 1.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.45 - 7.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.59T + 47T^{2} \)
53 \( 1 + (3.81 + 2.20i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 + 4.82iT - 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + 3.14iT - 71T^{2} \)
73 \( 1 + (4.62 + 2.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.475 + 0.823i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.34 + 1.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.31300288881629517104902564053, −10.77397957380578729095789453562, −9.916863447374161674226789468474, −8.917086905567430227148561994012, −8.166057124070707967313360849282, −6.27323436586592014320308385599, −4.81306398416806985191913133198, −3.79265394871713889110580446214, −3.19027004500515129461227614460, −1.22412927669598256306062945400, 1.59695443752278677657469476986, 4.08903770854515007083044018750, 5.35494846800254987921934399798, 6.16840255464923340107624179224, 7.13396513170093891443810126292, 7.82051194251738214279383744947, 8.813421602385688582269757284963, 9.256392817474150582921560965539, 11.52724622493173478618444702560, 11.77487105979862085084927924150

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