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Properties

Label 2-315-63.16-c1-0-30
Degree $2$
Conductor $315$
Sign $-0.251 - 0.967i$
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  + (−1.16 − 2.01i)2-s + (0.434 − 1.67i)3-s + (−1.70 + 2.95i)4-s − 5-s + (−3.88 + 1.07i)6-s + (0.459 − 2.60i)7-s + 3.27·8-s + (−2.62 − 1.45i)9-s + (1.16 + 2.01i)10-s − 2.95·11-s + (4.20 + 4.13i)12-s + (1.00 + 1.73i)13-s + (−5.78 + 2.10i)14-s + (−0.434 + 1.67i)15-s + (−0.399 − 0.692i)16-s + (−1.98 − 3.43i)17-s + ⋯
L(s)  = 1  + (−0.822 − 1.42i)2-s + (0.250 − 0.968i)3-s + (−0.852 + 1.47i)4-s − 0.447·5-s + (−1.58 + 0.438i)6-s + (0.173 − 0.984i)7-s + 1.15·8-s + (−0.874 − 0.485i)9-s + (0.367 + 0.636i)10-s − 0.891·11-s + (1.21 + 1.19i)12-s + (0.278 + 0.482i)13-s + (−1.54 + 0.562i)14-s + (−0.112 + 0.432i)15-s + (−0.0999 − 0.173i)16-s + (−0.480 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.251 - 0.967i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295412 + 0.382085i\)
\(L(\frac12)\) \(\approx\) \(0.295412 + 0.382085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.434 + 1.67i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.459 + 2.60i)T \)
good2 \( 1 + (1.16 + 2.01i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.98 + 3.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.55 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.433T + 23T^{2} \)
29 \( 1 + (-1.68 + 2.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.53 + 7.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0400 + 0.0693i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.435 + 0.753i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.92 - 3.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.24 + 7.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.89 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.50 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.54 + 7.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.95T + 71T^{2} \)
73 \( 1 + (5.84 + 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.15 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.126 - 0.218i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.58 + 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.67 + 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11266954500661371079085912588, −10.23006092918291040904353642007, −9.225550593364831336418140156980, −8.117192773253436144704193692796, −7.67520206272476250087557358372, −6.34667216812912120831206710793, −4.35976735256884320234380776430, −3.12918562780729154656166667336, −1.91065880510159091120445701801, −0.43128038305726193264965660009, 2.89810026093654108943090497476, 4.69228866616284836261573424368, 5.49818710614930704578629218620, 6.55078519315276398171926594476, 7.87692347019397264130450973556, 8.582049655932307563208284171282, 9.027011331921997059439023565405, 10.26657610019946989287946840541, 10.93188169391234173302440038120, 12.26151952411614349645040214839

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