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Properties

Label 2-315-63.4-c1-0-8
Degree $2$
Conductor $315$
Sign $0.685 - 0.728i$
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  + (−0.0127 + 0.0221i)2-s + (−1.71 + 0.211i)3-s + (0.999 + 1.73i)4-s + 5-s + (0.0172 − 0.0407i)6-s + (0.170 − 2.64i)7-s − 0.102·8-s + (2.91 − 0.728i)9-s + (−0.0127 + 0.0221i)10-s + 3.78·11-s + (−2.08 − 2.76i)12-s + (−2.77 + 4.79i)13-s + (0.0563 + 0.0375i)14-s + (−1.71 + 0.211i)15-s + (−1.99 + 3.46i)16-s + (−0.271 + 0.471i)17-s + ⋯
L(s)  = 1  + (−0.00904 + 0.0156i)2-s + (−0.992 + 0.122i)3-s + (0.499 + 0.865i)4-s + 0.447·5-s + (0.00706 − 0.0166i)6-s + (0.0642 − 0.997i)7-s − 0.0361·8-s + (0.970 − 0.242i)9-s + (−0.00404 + 0.00700i)10-s + 1.13·11-s + (−0.601 − 0.798i)12-s + (−0.768 + 1.33i)13-s + (0.0150 + 0.0100i)14-s + (−0.443 + 0.0546i)15-s + (−0.499 + 0.865i)16-s + (−0.0659 + 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.685 - 0.728i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.685 - 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10506 + 0.477417i\)
\(L(\frac12)\) \(\approx\) \(1.10506 + 0.477417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.211i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.170 + 2.64i)T \)
good2 \( 1 + (0.0127 - 0.0221i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 + (2.77 - 4.79i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.271 - 0.471i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.25T + 23T^{2} \)
29 \( 1 + (4.04 + 7.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.870 + 1.50i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.67 - 2.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.238 + 0.412i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.279 + 0.483i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.28 + 7.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.26 - 7.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.704 + 1.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.877 - 1.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.19 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-2.91 + 5.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.48 - 2.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.97 + 6.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.02 + 15.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.81 + 6.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.71190174518082372071186857311, −11.10925742710487038038595984318, −9.979509851042817666309830099242, −9.186445130809084199492680699975, −7.58190964632671369136639834894, −6.93647579573809669146092571703, −6.09753055362201302750772345693, −4.56247623236809878357950636922, −3.69470188472943298571161291540, −1.62900751552092645251802585092, 1.15732041926049737830053143442, 2.73935748605829544592677325850, 5.07759006017478520899429441287, 5.42916134226525032139528806446, 6.55862659361056356223140728510, 7.28077814697651210580994818983, 9.110917430932175135232532830041, 9.679422323956414314201787657742, 10.87658582159123454351049418950, 11.34551195638342570064408128705

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