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Properties

Label 2-315-63.4-c1-0-9
Degree $2$
Conductor $315$
Sign $-0.320 - 0.947i$
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  + (−0.349 + 0.605i)2-s + (1.42 + 0.990i)3-s + (0.755 + 1.30i)4-s + 5-s + (−1.09 + 0.514i)6-s + (−1.62 + 2.08i)7-s − 2.45·8-s + (1.03 + 2.81i)9-s + (−0.349 + 0.605i)10-s − 3.83·11-s + (−0.222 + 2.60i)12-s + (2.28 − 3.94i)13-s + (−0.695 − 1.71i)14-s + (1.42 + 0.990i)15-s + (−0.653 + 1.13i)16-s + (2.93 − 5.07i)17-s + ⋯
L(s)  = 1  + (−0.247 + 0.428i)2-s + (0.820 + 0.571i)3-s + (0.377 + 0.654i)4-s + 0.447·5-s + (−0.447 + 0.209i)6-s + (−0.614 + 0.789i)7-s − 0.867·8-s + (0.346 + 0.938i)9-s + (−0.110 + 0.191i)10-s − 1.15·11-s + (−0.0642 + 0.752i)12-s + (0.632 − 1.09i)13-s + (−0.185 − 0.457i)14-s + (0.366 + 0.255i)15-s + (−0.163 + 0.282i)16-s + (0.711 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913212 + 1.27357i\)
\(L(\frac12)\) \(\approx\) \(0.913212 + 1.27357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 - 0.990i)T \)
5 \( 1 - T \)
7 \( 1 + (1.62 - 2.08i)T \)
good2 \( 1 + (0.349 - 0.605i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 3.83T + 11T^{2} \)
13 \( 1 + (-2.28 + 3.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.93 + 5.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.386 - 0.670i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.90T + 23T^{2} \)
29 \( 1 + (3.95 + 6.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.01 - 3.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.23 - 7.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.60 - 2.78i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.19 - 7.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.66 + 2.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.32 + 2.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.36 - 5.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.50 - 6.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.28 + 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + (-7.00 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.69 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.442 - 0.766i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.950 - 1.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.07 + 5.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.00185004088918463008913616089, −10.82242272414564363234779151335, −9.826381309694220537967591228205, −9.065586069758773333333493524191, −8.139187826294466684629883079755, −7.45813480159193934176381443649, −6.05854434550565976671429328416, −5.05975503242564534189102221530, −3.12238307715981306838721632438, −2.76050607978228183806251601049, 1.24574914844423436132601456212, 2.52776929168521277128186008132, 3.75366376040744047575231215315, 5.61668325122934101990393146796, 6.64456585735579393603744053166, 7.44426509297991891837419276644, 8.791762136540144251490101606271, 9.519255656534726574200322043573, 10.45231045037883579290377965509, 11.09650032431564869264045185232

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