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Properties

Label 2-315-9.4-c1-0-18
Degree $2$
Conductor $315$
Sign $-0.833 + 0.552i$
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  + (−0.159 − 0.275i)2-s + (−1.72 − 0.0977i)3-s + (0.949 − 1.64i)4-s + (−0.5 + 0.866i)5-s + (0.248 + 0.492i)6-s + (−0.5 − 0.866i)7-s − 1.24·8-s + (2.98 + 0.338i)9-s + 0.318·10-s + (−1.66 − 2.87i)11-s + (−1.80 + 2.75i)12-s + (0.709 − 1.22i)13-s + (−0.159 + 0.275i)14-s + (0.949 − 1.44i)15-s + (−1.70 − 2.94i)16-s − 7.75·17-s + ⋯
L(s)  = 1  + (−0.112 − 0.195i)2-s + (−0.998 − 0.0564i)3-s + (0.474 − 0.822i)4-s + (−0.223 + 0.387i)5-s + (0.101 + 0.201i)6-s + (−0.188 − 0.327i)7-s − 0.438·8-s + (0.993 + 0.112i)9-s + 0.100·10-s + (−0.501 − 0.868i)11-s + (−0.520 + 0.794i)12-s + (0.196 − 0.341i)13-s + (−0.0425 + 0.0737i)14-s + (0.245 − 0.374i)15-s + (−0.425 − 0.736i)16-s − 1.88·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167760 - 0.556877i\)
\(L(\frac12)\) \(\approx\) \(0.167760 - 0.556877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.0977i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.159 + 0.275i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.66 + 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.709 + 1.22i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.75T + 17T^{2} \)
19 \( 1 + 3.18T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.270 - 0.468i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.85 + 6.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.03T + 37T^{2} \)
41 \( 1 + (0.0869 - 0.150i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.62 + 9.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.11 - 7.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (-3.90 + 6.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.67 - 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.61 + 2.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.03T + 71T^{2} \)
73 \( 1 - 6.45T + 73T^{2} \)
79 \( 1 + (-4.10 - 7.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.57 + 2.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.68T + 89T^{2} \)
97 \( 1 + (-6.87 - 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.97297105661759259837430573971, −10.76479245899764017142561626090, −9.786599683945149816151408716292, −8.467404345950946916944894467394, −7.03346330650307016571062455953, −6.37634870410465395499402693228, −5.48384574994580766568513418190, −4.18438924126699975112186986641, −2.37849875188317695733427604569, −0.46024807848489737060152572804, 2.22143986745155039627365848920, 4.05281036420426208006954178296, 4.99729064009907202609802584450, 6.46266004480581849705182730466, 6.96807781489943056730193483967, 8.213160217594797564394117110933, 9.124884726991468937355125856776, 10.36690055615545034641120633655, 11.29877890572054325410111914138, 11.97060402824447853446920367939

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