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Properties

Label 2-315-9.7-c1-0-19
Degree $2$
Conductor $315$
Sign $0.0348 + 0.999i$
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  + (0.669 − 1.15i)2-s + (0.704 − 1.58i)3-s + (0.104 + 0.181i)4-s + (0.5 + 0.866i)5-s + (−1.36 − 1.87i)6-s + (0.5 − 0.866i)7-s + 2.95·8-s + (−2.00 − 2.22i)9-s + 1.33·10-s + (−0.330 + 0.573i)11-s + (0.360 − 0.0378i)12-s + (−0.891 − 1.54i)13-s + (−0.669 − 1.15i)14-s + (1.72 − 0.181i)15-s + (1.76 − 3.06i)16-s + 1.33·17-s + ⋯
L(s)  = 1  + (0.473 − 0.819i)2-s + (0.406 − 0.913i)3-s + (0.0522 + 0.0905i)4-s + (0.223 + 0.387i)5-s + (−0.556 − 0.765i)6-s + (0.188 − 0.327i)7-s + 1.04·8-s + (−0.669 − 0.743i)9-s + 0.423·10-s + (−0.0997 + 0.172i)11-s + (0.103 − 0.0109i)12-s + (−0.247 − 0.428i)13-s + (−0.178 − 0.309i)14-s + (0.444 − 0.0467i)15-s + (0.442 − 0.766i)16-s + 0.324·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0348 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0348 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0348 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48591 - 1.43492i\)
\(L(\frac12)\) \(\approx\) \(1.48591 - 1.43492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.704 + 1.58i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.669 + 1.15i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.330 - 0.573i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.891 + 1.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (-2.60 - 4.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.46 - 4.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.344 + 0.596i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.53T + 37T^{2} \)
41 \( 1 + (-0.409 - 0.709i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.819 + 1.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + (2.74 + 4.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.87 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.30 - 5.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.79T + 71T^{2} \)
73 \( 1 - 0.0885T + 73T^{2} \)
79 \( 1 + (5.78 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.978 - 1.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (-8.33 + 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.50216267102918166936871136036, −10.82227931342818616530129416957, −9.769306849451440617498447856778, −8.439285045243587770624052742354, −7.50269678356061745572382803126, −6.79145105239507690325113058056, −5.33735434345723527259460223881, −3.79307443130603021475709983768, −2.78352596609215494262580098601, −1.62258224741643344565558709093, 2.23232450793004479345777780143, 4.02743460985799548239031572868, 4.96664802270684126937228166038, 5.75327852843245957311799671378, 6.91452699803415844288556194963, 8.184692234342873087706409351442, 8.922935831880875006607263527187, 10.06221784603670694960473025301, 10.74530672639384156394451878549, 11.85475954073590842997893471532

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