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Properties

Label 2-245-245.219-c1-0-8
Degree $2$
Conductor $245$
Sign $0.924 + 0.380i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (−0.348 − 0.0261i)2-s + (−0.132 − 0.428i)3-s + (−1.85 − 0.279i)4-s + (0.604 + 2.15i)5-s + (0.0349 + 0.152i)6-s + (0.493 − 2.59i)7-s + (1.32 + 0.301i)8-s + (2.31 − 1.57i)9-s + (−0.154 − 0.766i)10-s + (3.52 + 2.40i)11-s + (0.125 + 0.833i)12-s + (−1.69 − 3.52i)13-s + (−0.239 + 0.893i)14-s + (0.843 − 0.543i)15-s + (3.13 + 0.967i)16-s + (5.27 + 2.06i)17-s + ⋯
L(s)  = 1  + (−0.246 − 0.0184i)2-s + (−0.0763 − 0.247i)3-s + (−0.928 − 0.139i)4-s + (0.270 + 0.962i)5-s + (0.0142 + 0.0624i)6-s + (0.186 − 0.982i)7-s + (0.467 + 0.106i)8-s + (0.770 − 0.525i)9-s + (−0.0488 − 0.242i)10-s + (1.06 + 0.725i)11-s + (0.0362 + 0.240i)12-s + (−0.470 − 0.977i)13-s + (−0.0641 + 0.238i)14-s + (0.217 − 0.140i)15-s + (0.784 + 0.241i)16-s + (1.27 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.924 + 0.380i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01743 - 0.201079i\)
\(L(\frac12)\) \(\approx\) \(1.01743 - 0.201079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.604 - 2.15i)T \)
7 \( 1 + (-0.493 + 2.59i)T \)
good2 \( 1 + (0.348 + 0.0261i)T + (1.97 + 0.298i)T^{2} \)
3 \( 1 + (0.132 + 0.428i)T + (-2.47 + 1.68i)T^{2} \)
11 \( 1 + (-3.52 - 2.40i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (1.69 + 3.52i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-5.27 - 2.06i)T + (12.4 + 11.5i)T^{2} \)
19 \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.80 + 0.707i)T + (16.8 - 15.6i)T^{2} \)
29 \( 1 + (4.46 + 5.59i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (0.331 + 0.574i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.39 - 9.24i)T + (-35.3 + 10.9i)T^{2} \)
41 \( 1 + (0.619 - 2.71i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-10.2 + 2.33i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (8.68 + 0.650i)T + (46.4 + 7.00i)T^{2} \)
53 \( 1 + (1.81 - 12.0i)T + (-50.6 - 15.6i)T^{2} \)
59 \( 1 + (-0.363 + 0.337i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (7.38 - 1.11i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-5.51 + 3.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.06 + 7.60i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (9.86 - 0.739i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (-0.203 + 0.352i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.991 + 2.05i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (10.6 - 7.27i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + 0.902iT - 97T^{2} \)
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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

−12.12394596525293002660987887779, −10.82713916266728583791154981818, −9.895724699247712182576515633082, −9.583152016202436730833519954574, −7.85755801593539812334804697773, −7.22844263849818052616478926656, −6.06072373445150862091377865468, −4.54626346521407076021459094121, −3.49923331074323880439822729154, −1.23301130470399224464653423424, 1.49487577589942090951118178248, 3.80569458114183821813100932815, 4.92994359035916784347690986938, 5.68445018580745920729771402433, 7.43135775829808756818125204459, 8.544493744282995098988418637499, 9.324443223969455815547613664416, 9.741335285564527611516161311658, 11.29709249179038018005524516704, 12.31477041501791675566558473667

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