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Properties

Label 2-315-5.2-c2-0-10
Degree $2$
Conductor $315$
Sign $-0.939 + 0.341i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.41 − 2.41i)2-s + 7.68i·4-s + (−4.18 + 2.74i)5-s + (−1.87 − 1.87i)7-s + (8.90 − 8.90i)8-s + (16.7 + 3.47i)10-s + 20.9·11-s + (−9.34 + 9.34i)13-s + 9.04i·14-s − 12.2·16-s + (−7.08 − 7.08i)17-s − 14.9i·19-s + (−21.0 − 32.1i)20-s + (−50.5 − 50.5i)22-s + (−12.9 + 12.9i)23-s + ⋯
L(s)  = 1  + (−1.20 − 1.20i)2-s + 1.92i·4-s + (−0.836 + 0.548i)5-s + (−0.267 − 0.267i)7-s + (1.11 − 1.11i)8-s + (1.67 + 0.347i)10-s + 1.90·11-s + (−0.718 + 0.718i)13-s + 0.645i·14-s − 0.768·16-s + (−0.416 − 0.416i)17-s − 0.786i·19-s + (−1.05 − 1.60i)20-s + (−2.29 − 2.29i)22-s + (−0.563 + 0.563i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.939 + 0.341i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.939 + 0.341i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0747046 - 0.424291i\)
\(L(\frac12)\) \(\approx\) \(0.0747046 - 0.424291i\)
\(L(2)\) 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 \( 1 + (4.18 - 2.74i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (2.41 + 2.41i)T + 4iT^{2} \)
11 \( 1 - 20.9T + 121T^{2} \)
13 \( 1 + (9.34 - 9.34i)T - 169iT^{2} \)
17 \( 1 + (7.08 + 7.08i)T + 289iT^{2} \)
19 \( 1 + 14.9iT - 361T^{2} \)
23 \( 1 + (12.9 - 12.9i)T - 529iT^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 - 12.8T + 961T^{2} \)
37 \( 1 + (31.7 + 31.7i)T + 1.36e3iT^{2} \)
41 \( 1 + 69.4T + 1.68e3T^{2} \)
43 \( 1 + (-4.46 + 4.46i)T - 1.84e3iT^{2} \)
47 \( 1 + (-4.41 - 4.41i)T + 2.20e3iT^{2} \)
53 \( 1 + (-48.5 + 48.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 29.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.09T + 3.72e3T^{2} \)
67 \( 1 + (1.39 + 1.39i)T + 4.48e3iT^{2} \)
71 \( 1 - 15.9T + 5.04e3T^{2} \)
73 \( 1 + (-32.4 + 32.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (-83.6 + 83.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 62.7iT - 7.92e3T^{2} \)
97 \( 1 + (85.4 + 85.4i)T + 9.40e3iT^{2} \)
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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

−11.13169896610009943365487025105, −10.02565687376742446386724128546, −9.339480114566140630310772944405, −8.525719178390926324394634124468, −7.33405436205312349154546943558, −6.63092041403650930097258624507, −4.28285767484503436542897310328, −3.42498630492843270799860321736, −2.02244299065993097683503096769, −0.35203383022357526404259933753, 1.23788292904727450811974387127, 3.76012820773545500635105182701, 5.17725087260925972731119867050, 6.40492150838625656702763311351, 7.10077884304109021445288380644, 8.249692099854971922404828996909, 8.743287122063030495544487447569, 9.628570467570770345105140648170, 10.55492283191052290599626183082, 11.92660244549081654684777139435

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