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Properties

Label 2-315-35.9-c1-0-12
Degree $2$
Conductor $315$
Sign $0.961 - 0.275i$
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 + (0.881 − 2.05i)5-s + (2.40 + 1.09i)7-s − 1.47i·8-s + (3.19 − 2.38i)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 + (0.394 − 0.919i)5-s + (0.910 + 0.413i)7-s − 0.520i·8-s + (1.00 − 0.754i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38174 + 0.334703i\)
\(L(\frac12)\) \(\approx\) \(2.38174 + 0.334703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.881 + 2.05i)T \)
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

−12.01475433386028684339907773892, −11.00229814321899318232108701568, −9.700109754957158785778078630219, −8.728416635177824332198747933503, −7.83986645681109962815217561029, −6.44906269559989640220814280151, −5.58160601588311628307222615464, −4.83666118582410779089738816177, −3.83347100535329966169608557824, −1.77036337582020940453794208988, 2.13926093655411811117567975190, 3.12456164902069352578207065540, 4.51843556111640836958351688216, 5.24995344004690354973428631724, 6.62434199381961520813513642749, 7.63350364784151696102087789690, 8.809136346001452786217328782085, 10.24727016500260276730213158742, 11.01471042936383847358138045582, 11.41011166812772224374117952681

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