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Properties

Label 2-315-63.5-c1-0-2
Degree $2$
Conductor $315$
Sign $0.999 + 0.0222i$
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  − 2.72i·2-s + (−1.55 + 0.754i)3-s − 5.44·4-s + (0.5 + 0.866i)5-s + (2.05 + 4.25i)6-s + (−2.63 + 0.200i)7-s + 9.39i·8-s + (1.86 − 2.35i)9-s + (2.36 − 1.36i)10-s + (0.246 + 0.142i)11-s + (8.48 − 4.10i)12-s + (2.89 + 1.67i)13-s + (0.545 + 7.19i)14-s + (−1.43 − 0.972i)15-s + 14.7·16-s + (2.65 + 4.60i)17-s + ⋯
L(s)  = 1  − 1.92i·2-s + (−0.900 + 0.435i)3-s − 2.72·4-s + (0.223 + 0.387i)5-s + (0.840 + 1.73i)6-s + (−0.997 + 0.0756i)7-s + 3.32i·8-s + (0.620 − 0.784i)9-s + (0.747 − 0.431i)10-s + (0.0741 + 0.0428i)11-s + (2.44 − 1.18i)12-s + (0.803 + 0.464i)13-s + (0.145 + 1.92i)14-s + (−0.369 − 0.251i)15-s + 3.68·16-s + (0.644 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0222i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.999 + 0.0222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507828 - 0.00565600i\)
\(L(\frac12)\) \(\approx\) \(0.507828 - 0.00565600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.55 - 0.754i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.63 - 0.200i)T \)
good2 \( 1 + 2.72iT - 2T^{2} \)
11 \( 1 + (-0.246 - 0.142i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.89 - 1.67i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.65 - 4.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.743 - 0.429i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.85 - 3.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.98 - 1.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.14iT - 31T^{2} \)
37 \( 1 + (4.53 - 7.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.223 - 0.387i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.01T + 47T^{2} \)
53 \( 1 + (0.483 - 0.279i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.19T + 59T^{2} \)
61 \( 1 + 8.04iT - 61T^{2} \)
67 \( 1 - 1.60T + 67T^{2} \)
71 \( 1 - 8.30iT - 71T^{2} \)
73 \( 1 + (-3.55 + 2.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.58T + 79T^{2} \)
83 \( 1 + (2.29 + 3.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.97 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.69 + 2.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.59350334908034749341826017888, −10.78145751741351162136785791034, −9.926417826320735712294710877089, −9.638295145172029296466451660309, −8.345446035225329739496564088040, −6.40129889583502017827482082424, −5.43731218321484756434392769276, −3.98334393529397148202288435925, −3.36828215867360585581330771368, −1.57171809538035224806243211583, 0.44075525466636388592164262445, 3.91907321149926349177537789650, 5.24641006218022675724992123085, 5.88463696975728757831619942983, 6.69407979263970578651038862116, 7.51691111496816334319011634445, 8.519582749258645613243568734988, 9.565631976287281881460928145068, 10.37283272987123507635952234162, 12.07377173852430201676826778247

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