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Properties

Label 2-315-63.41-c1-0-24
Degree $2$
Conductor $315$
Sign $-0.873 + 0.486i$
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  + (−1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (0.5 − 0.866i)5-s + (0.5 − 2.59i)7-s + (1.5 + 2.59i)9-s + (−3 + 1.73i)11-s + (3 − 1.73i)12-s + (−4.5 − 2.59i)13-s + (−1.5 + 0.866i)15-s + (−1.99 − 3.46i)16-s − 3·17-s − 3.46i·19-s + (0.999 + 1.73i)20-s + (−3 + 3.46i)21-s + (−6 − 3.46i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.223 − 0.387i)5-s + (0.188 − 0.981i)7-s + (0.5 + 0.866i)9-s + (−0.904 + 0.522i)11-s + (0.866 − 0.499i)12-s + (−1.24 − 0.720i)13-s + (−0.387 + 0.223i)15-s + (−0.499 − 0.866i)16-s − 0.727·17-s − 0.794i·19-s + (0.223 + 0.387i)20-s + (−0.654 + 0.755i)21-s + (−1.25 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0838728 - 0.323072i\)
\(L(\frac12)\) \(\approx\) \(0.0838728 - 0.323072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.46iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (6 - 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.39502071496164773830388195352, −10.36991115359669123604800749312, −9.567634371287389999357717214378, −7.943463777893820634078176771628, −7.66183269479663247546267361554, −6.47636871994357806988692688867, −4.93082024653125411351835608108, −4.47472094678161490026605803293, −2.48414006759323295599288425819, −0.25414839763193144939399781602, 2.17506326725106952223255566359, 4.15254855460881646394896294953, 5.37501960139993051197869037323, 5.75152316705375807202607760442, 6.96294748897275320710413980460, 8.514538950817179259346087612720, 9.573330021712830088844340274017, 10.11882182613055785969880292227, 11.05671761065448445732728550764, 11.86131838813587674782220722380

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