Properties

Label 2-315-63.38-c1-0-7
Degree $2$
Conductor $315$
Sign $-0.413 - 0.910i$
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  + 0.645i·2-s + (0.803 + 1.53i)3-s + 1.58·4-s + (−0.5 + 0.866i)5-s + (−0.991 + 0.519i)6-s + (−2.64 + 0.0866i)7-s + 2.31i·8-s + (−1.70 + 2.46i)9-s + (−0.559 − 0.322i)10-s + (−1.35 + 0.781i)11-s + (1.27 + 2.42i)12-s + (0.956 − 0.552i)13-s + (−0.0559 − 1.70i)14-s + (−1.73 − 0.0712i)15-s + 1.67·16-s + (0.145 − 0.251i)17-s + ⋯
L(s)  = 1  + 0.456i·2-s + (0.463 + 0.885i)3-s + 0.791·4-s + (−0.223 + 0.387i)5-s + (−0.404 + 0.211i)6-s + (−0.999 + 0.0327i)7-s + 0.818i·8-s + (−0.569 + 0.822i)9-s + (−0.176 − 0.102i)10-s + (−0.408 + 0.235i)11-s + (0.367 + 0.701i)12-s + (0.265 − 0.153i)13-s + (−0.0149 − 0.456i)14-s + (−0.446 − 0.0183i)15-s + 0.417·16-s + (0.0352 − 0.0610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.413 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846849 + 1.31466i\)
\(L(\frac12)\) \(\approx\) \(0.846849 + 1.31466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.803 - 1.53i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.64 - 0.0866i)T \)
good2 \( 1 - 0.645iT - 2T^{2} \)
11 \( 1 + (1.35 - 0.781i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.956 + 0.552i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.145 + 0.251i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.30 + 3.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.59 - 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.416 + 0.240i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5iT - 31T^{2} \)
37 \( 1 + (-2.72 - 4.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.348 - 0.603i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.52 - 2.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.306T + 47T^{2} \)
53 \( 1 + (7.22 + 4.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.52T + 59T^{2} \)
61 \( 1 + 2.24iT - 61T^{2} \)
67 \( 1 - 7.93T + 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + (10.9 + 6.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.52T + 79T^{2} \)
83 \( 1 + (-0.398 + 0.690i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.51 - 9.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.45 + 4.30i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.61004325273198798469772365579, −11.04762035486661419306749210927, −9.981827446260356025287492328508, −9.301249175097048710253477515178, −7.992522629904455170238135065714, −7.22616596363220923163442787801, −6.11274142853816579354163911332, −5.04759182313052619461469863578, −3.43940960475841512421206811846, −2.65366759054511820448848286016, 1.15541897150179878014735480528, 2.76021523516172531401976178901, 3.56774742203130172510791542678, 5.61224744191128011893054314265, 6.70276700675286964766006246762, 7.37452024244549313870306636636, 8.512543124678951481239691649246, 9.489387662061040942969915974509, 10.53775112526092877158868697716, 11.55100336917780669521445868120

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