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Properties

Label 2-315-63.5-c1-0-27
Degree $2$
Conductor $315$
Sign $-0.929 - 0.368i$
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  − 2.51i·2-s + (−0.170 + 1.72i)3-s − 4.34·4-s + (−0.5 − 0.866i)5-s + (4.34 + 0.430i)6-s + (1.80 − 1.93i)7-s + 5.90i·8-s + (−2.94 − 0.588i)9-s + (−2.18 + 1.25i)10-s + (−5.56 − 3.21i)11-s + (0.742 − 7.48i)12-s + (−1.77 − 1.02i)13-s + (−4.86 − 4.55i)14-s + (1.57 − 0.713i)15-s + 6.18·16-s + (−1.68 − 2.92i)17-s + ⋯
L(s)  = 1  − 1.78i·2-s + (−0.0986 + 0.995i)3-s − 2.17·4-s + (−0.223 − 0.387i)5-s + (1.77 + 0.175i)6-s + (0.682 − 0.730i)7-s + 2.08i·8-s + (−0.980 − 0.196i)9-s + (−0.689 + 0.398i)10-s + (−1.67 − 0.969i)11-s + (0.214 − 2.16i)12-s + (−0.491 − 0.283i)13-s + (−1.30 − 1.21i)14-s + (0.407 − 0.184i)15-s + 1.54·16-s + (−0.409 − 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128115 + 0.669921i\)
\(L(\frac12)\) \(\approx\) \(0.128115 + 0.669921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.170 - 1.72i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.80 + 1.93i)T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
11 \( 1 + (5.56 + 3.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.77 + 1.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.68 + 2.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.51 - 2.02i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.48 - 1.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.49 + 3.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.965iT - 31T^{2} \)
37 \( 1 + (1.31 - 2.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.50 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.788 - 1.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.05T + 47T^{2} \)
53 \( 1 + (3.22 - 1.86i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 7.33iT - 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 5.13iT - 71T^{2} \)
73 \( 1 + (9.61 - 5.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.85T + 79T^{2} \)
83 \( 1 + (-0.549 - 0.951i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.423 - 0.733i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.96 - 1.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.19428515596925884212790327080, −10.27273100522643904375101621485, −9.843148363577533472760498126635, −8.591588584273798615299082561042, −7.86349629336288073633413608549, −5.38090755631148007335796954106, −4.74955661897024946328890193343, −3.64926904681790041838061204384, −2.60176859444445946475042062562, −0.48789230235841854425808979930, 2.43697034087067646846117490490, 4.76859934751154887421737269743, 5.45741087091914804440195110616, 6.56055216970086454455965697302, 7.39838859129958067578067403011, 8.004666831245109070200048067206, 8.740459110041245248959055661513, 10.09565350289232375825299730933, 11.42975595511912758405087537922, 12.54147602504398768460348049128

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