Properties

Label 2-145-145.109-c1-0-11
Degree $2$
Conductor $145$
Sign $-0.236 - 0.971i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  + (−0.584 − 2.55i)2-s + (−1.84 − 0.890i)3-s + (−4.40 + 2.12i)4-s + (0.258 − 2.22i)5-s + (−1.19 + 5.25i)6-s + (−0.0881 + 0.183i)7-s + (4.73 + 5.94i)8-s + (0.753 + 0.944i)9-s + (−5.83 + 0.636i)10-s + (2.83 + 2.26i)11-s + 10.0·12-s + (−1.95 − 1.55i)13-s + (0.520 + 0.118i)14-s + (−2.45 + 3.87i)15-s + (6.33 − 7.94i)16-s − 6.11·17-s + ⋯
L(s)  = 1  + (−0.413 − 1.81i)2-s + (−1.06 − 0.513i)3-s + (−2.20 + 1.06i)4-s + (0.115 − 0.993i)5-s + (−0.489 + 2.14i)6-s + (−0.0333 + 0.0691i)7-s + (1.67 + 2.10i)8-s + (0.251 + 0.314i)9-s + (−1.84 + 0.201i)10-s + (0.854 + 0.681i)11-s + 2.89·12-s + (−0.542 − 0.432i)13-s + (0.138 + 0.0317i)14-s + (−0.633 + 1.00i)15-s + (1.58 − 1.98i)16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.236 - 0.971i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ -0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240716 + 0.306434i\)
\(L(\frac12)\) \(\approx\) \(0.240716 + 0.306434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.258 + 2.22i)T \)
29 \( 1 + (1.18 + 5.25i)T \)
good2 \( 1 + (0.584 + 2.55i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (1.84 + 0.890i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (0.0881 - 0.183i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (-2.83 - 2.26i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.95 + 1.55i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + 6.11T + 17T^{2} \)
19 \( 1 + (2.49 + 5.17i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-5.97 - 1.36i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (5.46 - 1.24i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (1.60 + 2.00i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 8.88iT - 41T^{2} \)
43 \( 1 + (-1.31 + 5.75i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.0655 - 0.0822i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-5.87 + 1.34i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 6.10T + 59T^{2} \)
61 \( 1 + (1.48 - 3.08i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-7.05 + 5.62i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-1.93 + 2.42i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.30 + 10.1i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (7.19 - 5.73i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-0.395 - 0.820i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.104 + 0.0237i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.72 - 2.27i)T + (60.4 - 75.8i)T^{2} \)
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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

−12.26294270640351282160941218010, −11.43585641764874641956829183010, −10.70400805453819288954122627543, −9.300493435915669835203662352346, −8.855134702851270301289482492289, −7.03631714484240044837801809469, −5.24671620136453526578077554919, −4.20388585030016644737561550345, −2.08994110835999398192396598542, −0.51037114207300671892731220161, 4.20790805348815189029537416132, 5.46331816954729168894523357267, 6.43972222953316526658484356431, 6.98852775457190402221592927430, 8.478516261589258873964541936370, 9.532758551911785748447537241536, 10.61323214309627869906874971830, 11.44754174051458592634008509294, 13.18269362450471906034909758165, 14.38672498722161407506860776374

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