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Properties

Label 2-315-63.16-c1-0-11
Degree $2$
Conductor $315$
Sign $-0.995 + 0.0937i$
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  + (1.32 + 2.29i)2-s + (−0.540 + 1.64i)3-s + (−2.51 + 4.35i)4-s + 5-s + (−4.49 + 0.941i)6-s + (2.58 + 0.545i)7-s − 8.04·8-s + (−2.41 − 1.77i)9-s + (1.32 + 2.29i)10-s + 5.32·11-s + (−5.81 − 6.49i)12-s + (−2.30 − 3.98i)13-s + (2.17 + 6.67i)14-s + (−0.540 + 1.64i)15-s + (−5.63 − 9.76i)16-s + (−0.923 − 1.60i)17-s + ⋯
L(s)  = 1  + (0.937 + 1.62i)2-s + (−0.311 + 0.950i)3-s + (−1.25 + 2.17i)4-s + 0.447·5-s + (−1.83 + 0.384i)6-s + (0.978 + 0.206i)7-s − 2.84·8-s + (−0.805 − 0.592i)9-s + (0.419 + 0.726i)10-s + 1.60·11-s + (−1.67 − 1.87i)12-s + (−0.638 − 1.10i)13-s + (0.582 + 1.78i)14-s + (−0.139 + 0.424i)15-s + (−1.40 − 2.44i)16-s + (−0.224 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.995 + 0.0937i$
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: \((2,\ 315,\ (\ :1/2),\ -0.995 + 0.0937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0937005 - 1.99542i\)
\(L(\frac12)\) \(\approx\) \(0.0937005 - 1.99542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.540 - 1.64i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.58 - 0.545i)T \)
good2 \( 1 + (-1.32 - 2.29i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 - 5.32T + 11T^{2} \)
13 \( 1 + (2.30 + 3.98i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.923 + 1.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + (-2.11 + 3.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.84 - 3.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.06 - 8.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.59 + 2.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.34 - 7.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.35 + 9.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.74 - 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.965 + 1.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.44 - 2.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.828 - 1.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.79T + 71T^{2} \)
73 \( 1 + (1.45 + 2.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.32 + 7.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.26 + 3.91i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.12 + 5.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.38 - 7.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.12236764952055987610727552236, −11.58705329240606017954736426858, −10.06982240784564566313878360006, −8.998227831375951880643137489776, −8.284864899738810281274026983644, −7.02223431257893131804065387833, −6.08247611383651548050919342220, −5.13869958569823745545842614650, −4.55328092552377398162144681641, −3.28777140880642229046403951919, 1.42391578517901143070875656146, 2.07400168128440036956348783981, 3.82260860205493157149063247116, 4.87263492621564121325344898310, 5.93947672818350073212309656171, 7.00749369055212926750150577121, 8.667099533440966581239492085141, 9.591346187366583368714515962632, 10.73336640766548988676704180867, 11.52782623079047832635938943926

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