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Properties

Label 2-315-63.38-c1-0-26
Degree $2$
Conductor $315$
Sign $0.776 + 0.630i$
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  − 0.0264i·2-s + (1.26 − 1.18i)3-s + 1.99·4-s + (0.5 − 0.866i)5-s + (−0.0313 − 0.0333i)6-s + (2.64 + 0.130i)7-s − 0.105i·8-s + (0.183 − 2.99i)9-s + (−0.0228 − 0.0132i)10-s + (−5.47 + 3.15i)11-s + (2.52 − 2.37i)12-s + (−2.98 + 1.72i)13-s + (0.00345 − 0.0697i)14-s + (−0.396 − 1.68i)15-s + 3.99·16-s + (−1.13 + 1.96i)17-s + ⋯
L(s)  = 1  − 0.0186i·2-s + (0.728 − 0.685i)3-s + 0.999·4-s + (0.223 − 0.387i)5-s + (−0.0127 − 0.0135i)6-s + (0.998 + 0.0494i)7-s − 0.0373i·8-s + (0.0610 − 0.998i)9-s + (−0.00723 − 0.00417i)10-s + (−1.64 + 0.952i)11-s + (0.728 − 0.684i)12-s + (−0.828 + 0.478i)13-s + (0.000923 − 0.0186i)14-s + (−0.102 − 0.435i)15-s + 0.998·16-s + (−0.274 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.776 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90626 - 0.676418i\)
\(L(\frac12)\) \(\approx\) \(1.90626 - 0.676418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.26 + 1.18i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.64 - 0.130i)T \)
good2 \( 1 + 0.0264iT - 2T^{2} \)
11 \( 1 + (5.47 - 3.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.98 - 1.72i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.13 - 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.32 - 2.49i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.546 - 0.315i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 0.713i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.96iT - 31T^{2} \)
37 \( 1 + (-4.69 - 8.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.66 + 8.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.21 - 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (2.53 + 1.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.273T + 59T^{2} \)
61 \( 1 + 5.90iT - 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 - 5.59iT - 71T^{2} \)
73 \( 1 + (0.0467 + 0.0270i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.96T + 79T^{2} \)
83 \( 1 + (4.66 - 8.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0707 + 0.122i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.37 - 0.791i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.74624680473098092118898129346, −10.61685298512290447837008805387, −9.766557522846048076648459650710, −8.360108899237461194299180286876, −7.78294158116845797890611119586, −6.96942433406533920289465684814, −5.71274404625264385255603482460, −4.41546083204991462451371160599, −2.48521690825039861928788657960, −1.88396631250719907026115959266, 2.32626247938817393949714439023, 2.99285546532870674704359540218, 4.75830782542521533433985264310, 5.66574041038172315832496357796, 7.21788851477738495947827477444, 7.938149889156321162401627126177, 8.787984448968416745921022589337, 10.30777782793770002350986481811, 10.65672483108948614661389701130, 11.38909563860336979585989479104

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