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Properties

Label 2-315-63.25-c1-0-30
Degree $2$
Conductor $315$
Sign $0.743 + 0.668i$
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  + 2.08·2-s + (0.878 − 1.49i)3-s + 2.35·4-s + (−0.5 + 0.866i)5-s + (1.83 − 3.11i)6-s + (−0.122 − 2.64i)7-s + 0.738·8-s + (−1.45 − 2.62i)9-s + (−1.04 + 1.80i)10-s + (0.00772 + 0.0133i)11-s + (2.06 − 3.51i)12-s + (3.01 + 5.21i)13-s + (−0.256 − 5.51i)14-s + (0.853 + 1.50i)15-s − 3.16·16-s + (−0.453 + 0.786i)17-s + ⋯
L(s)  = 1  + 1.47·2-s + (0.507 − 0.861i)3-s + 1.17·4-s + (−0.223 + 0.387i)5-s + (0.748 − 1.27i)6-s + (−0.0464 − 0.998i)7-s + 0.261·8-s + (−0.485 − 0.874i)9-s + (−0.329 + 0.571i)10-s + (0.00232 + 0.00403i)11-s + (0.597 − 1.01i)12-s + (0.834 + 1.44i)13-s + (−0.0685 − 1.47i)14-s + (0.220 + 0.389i)15-s − 0.791·16-s + (−0.110 + 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74082 - 1.05020i\)
\(L(\frac12)\) \(\approx\) \(2.74082 - 1.05020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.878 + 1.49i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.122 + 2.64i)T \)
good2 \( 1 - 2.08T + 2T^{2} \)
11 \( 1 + (-0.00772 - 0.0133i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.01 - 5.21i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.453 - 0.786i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.94 - 6.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.66 - 2.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.08 + 1.88i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + (4.79 + 8.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.46 + 9.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.234 + 0.406i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (1.39 - 2.41i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 - 8.25T + 61T^{2} \)
67 \( 1 - 5.32T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (0.466 - 0.808i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.02T + 79T^{2} \)
83 \( 1 + (8.41 - 14.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.33 + 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.11 + 5.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.85256522874396316369965751165, −11.12066730788958523042486378728, −9.746488179274350629122089701186, −8.470897763424224447648173492400, −7.29832816548523336509013712834, −6.64227581153168181281571571073, −5.66394647286111111626152087757, −3.96292657008901989201634622238, −3.55203982899111176119122576381, −1.84734896222510379551789419536, 2.74578640367610201126371926352, 3.44103644073510395097787692541, 4.87324794210073827746483314878, 5.26887861093535873610996796493, 6.47691724824531521020721710954, 8.161730987399052797329847161298, 8.864771458590517605359924195868, 9.949770729163885210350917528837, 11.20032057970582687502521526752, 11.83254692511695617516482318147

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