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Properties

Label 2-315-35.4-c1-0-0
Degree $2$
Conductor $315$
Sign $-0.999 - 0.0421i$
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  + (−1.54 + 0.890i)2-s + (0.587 − 1.01i)4-s + (1.33 + 1.79i)5-s + (−2.40 + 1.09i)7-s − 1.47i·8-s + (−3.66 − 1.56i)10-s + (−1.03 + 1.80i)11-s + 3.13i·13-s + (2.74 − 3.83i)14-s + (2.48 + 4.30i)16-s + (1.84 + 1.06i)17-s + (−3.86 − 6.70i)19-s + (2.60 − 0.310i)20-s − 3.70i·22-s + (−4.79 + 2.76i)23-s + ⋯
L(s)  = 1  + (−1.09 + 0.629i)2-s + (0.293 − 0.508i)4-s + (0.598 + 0.800i)5-s + (−0.910 + 0.413i)7-s − 0.520i·8-s + (−1.15 − 0.496i)10-s + (−0.313 + 0.542i)11-s + 0.868i·13-s + (0.732 − 1.02i)14-s + (0.621 + 1.07i)16-s + (0.447 + 0.258i)17-s + (−0.887 − 1.53i)19-s + (0.583 − 0.0694i)20-s − 0.789i·22-s + (−0.999 + 0.577i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0421i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.999 - 0.0421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00963251 + 0.456916i\)
\(L(\frac12)\) \(\approx\) \(0.00963251 + 0.456916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.33 - 1.79i)T \)
7 \( 1 + (2.40 - 1.09i)T \)
good2 \( 1 + (1.54 - 0.890i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.03 - 1.80i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
17 \( 1 + (-1.84 - 1.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.86 + 6.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.79 - 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 + (1.45 - 2.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.04 - 1.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.99T + 41T^{2} \)
43 \( 1 + 4.99iT - 43T^{2} \)
47 \( 1 + (-2.11 + 1.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.58 - 4.95i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.47 - 2.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.44 - 9.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.32 - 1.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + (-7.40 - 4.27i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.05 - 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.75iT - 83T^{2} \)
89 \( 1 + (0.309 + 0.535i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.296iT - 97T^{2} \)
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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

−12.07276321179358370829713119568, −10.78318163994874321811511190333, −9.940712198911055943007573813117, −9.351421126352855622439087470668, −8.481010921703974778080834446756, −7.07823508446245738878034381724, −6.75999124201427116955599212892, −5.61247047381589886426062511360, −3.75530596501403240531091948833, −2.21271662984883879796598298212, 0.44822226343589439504238509565, 2.04593538114357094026403856012, 3.59814423275813374928723537104, 5.32982797731706564809465360843, 6.20055531073859915331584902846, 7.88061458074925577140863401991, 8.479242426665482402622683318210, 9.587548371733024353527622868416, 10.10990018443458164042791500394, 10.78401890140873353641381795026

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