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Properties

Label 2-315-63.25-c1-0-1
Degree $2$
Conductor $315$
Sign $0.166 - 0.986i$
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  − 2.56·2-s + (−1.18 − 1.26i)3-s + 4.55·4-s + (0.5 − 0.866i)5-s + (3.02 + 3.24i)6-s + (−0.992 + 2.45i)7-s − 6.53·8-s + (−0.204 + 2.99i)9-s + (−1.28 + 2.21i)10-s + (−1.90 − 3.29i)11-s + (−5.38 − 5.76i)12-s + (−0.798 − 1.38i)13-s + (2.54 − 6.27i)14-s + (−1.68 + 0.390i)15-s + 7.63·16-s + (−1.84 + 3.18i)17-s + ⋯
L(s)  = 1  − 1.81·2-s + (−0.682 − 0.730i)3-s + 2.27·4-s + (0.223 − 0.387i)5-s + (1.23 + 1.32i)6-s + (−0.375 + 0.926i)7-s − 2.31·8-s + (−0.0682 + 0.997i)9-s + (−0.404 + 0.701i)10-s + (−0.574 − 0.994i)11-s + (−1.55 − 1.66i)12-s + (−0.221 − 0.383i)13-s + (0.678 − 1.67i)14-s + (−0.435 + 0.100i)15-s + 1.90·16-s + (−0.446 + 0.772i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.166 - 0.986i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174754 + 0.147742i\)
\(L(\frac12)\) \(\approx\) \(0.174754 + 0.147742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.992 - 2.45i)T \)
good2 \( 1 + 2.56T + 2T^{2} \)
11 \( 1 + (1.90 + 3.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.798 + 1.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.84 - 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.68 - 2.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.452 - 0.783i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.97 - 8.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 + (-3.90 - 6.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.38 - 11.0i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.849 + 1.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.68T + 47T^{2} \)
53 \( 1 + (1.61 - 2.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.25T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 + 3.66T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (1.47 - 2.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + (2.67 - 4.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.998 + 1.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.25 - 9.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.53508689551222199067522368400, −10.85310735553054692526983123105, −9.956695988432032302473144089927, −8.894501026343135106119049905180, −8.237503315331121218150099542587, −7.35064566085227997966877914193, −6.15063621458297440885915079284, −5.54172050275248114980877965264, −2.73733542862526917399819188795, −1.39127802116571179836336546741, 0.32624152506042884480634150169, 2.41303469599217005421944744229, 4.21937824528127784177858523110, 5.83052305133158053158601587916, 7.14191302158289694708473954578, 7.36946806537162870406831082875, 9.109410913122088242564102904011, 9.626922449702040472875472392529, 10.34483807263898876686988001478, 11.01479648648510251848586695873

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